Std12-BM-TM-2

tf fj

nkiy - ulh ML

lhik xU ghtbra

lhik xU bgUFw

lhik kjjikaw bra

j ehLj ehLj ehLj ehLj ehLghl fHfghl fHfghl fHfghl fHfghl fHff rhiy/ bridf rhiy/ bridf rhiy/ bridf rhiy/ bridf rhiy/ brid - 600 006.

bjhFbjhFbjhFbjhFbjhF -2

jehL muRKjg - 2005

,ulh g 2006

jiytjiytjiytjiytjiytKidt. r. mnjhuh,iznguha/ fjJiwkhy f/brid - 5.

nkyh -thsf-yhafnkyh -thsf-yhafnkyh -thsf-yhafnkyh -thsf-yhafnkyh -thsf-yhaf

U. ,uh. _njiy iuahsfjJiwkhy fbrid - 5.

U. e. unknjiy iuahsfjJiwmuR Mlt fiy fejd/ brid - 35.

yhafyhafyhafyhafyhafU. R. ,uhkruKJfiy gljh Majhngil nkiygjhngil/ brid-2.

U. rh. ,uhkKJfiy gljh Mab#anfhgh fnuhoah nja nkiyg/ HF jhgu/ brid-59.

U. r. nt. gkehgKJfiy gljh Ma,J nkiygUtnf/ brid-5.

U. ntQ. ufha iuahs (K..)khy fbrid - 5.

iy : %.iy : %.iy : %.iy : %.iy : %.ghlf jah :

jehL muRfhf g f ,aff/ jehL

, 60 .v.v. jh mlgLsJ.

nky h -t hs nky h -t hs nky h -t hs nky h -t hs nky h -t hs Kidt. kh.bu. thr,iznguha/ a Jiw/brid gfiyfHfbrid - 5.

ghl FGghl FGghl FGghl FGghl FG

Uk. K. khKJfiy gljh Mabg.R. nkiy g (ika)ikyh/ brid-4.

Uk. mk uh#hKJfiy gljh Maey Ma bk. nkiygfrhiy/ brid-6.

iii

Kfiu

``vj X cik f bjthd kW mHfhd TW,W fj totijna mila ntLpp - bjhu.

bghUaYfhd nehg gR bgwtf mWgJGfhoF nkgnlh fjJt bghUa _yKjyhdrhjidf br-jtf. mjifa bghUa tYef cafjij MJ gwnjhL mjid bgUbghUa kW fjbghUa Mat ca M-fSF btfukhf gagLd.

lh~nghL gfiy fHf Jiw nguha Kidtnfh vgtU bghUa tYd Kidt bkl vgtU,izJ 1970M ML/ fhyngh VgL khwij FFtifbfG rkghL Nu xiw fLoJbghUshjhubfd 1997M ML nehg gR bgwd. ,Nubjiy fhy/ iyf/ to j kW rij khW jikvw ehF khf mogil iyia khF tifmikUjJ. ,Nu eilKiw bgJ gaglnjhlyhk/mbkf gF rijiana khwkila br-jJ.

bghUa vgJ y btgil cikfis mogilahfbfhL jUf Kiwia gagL tUfgLtdtiw rhjma vW fUjglJ. Mdh ,W bghUa KYcUkhlJ. tiuglf/ rkghLf kW a MatVuhskhd gaghLf/ bghUa jikia khld. ykhf Jt gogoahf kw khfis F dmtilnaahd bjhlig/ kW bghUshjhu flik cmik jJtij Muha fj gagLjgLwJ. ,jkhfa bghUa cikfis fL mtiw bgUks gagLjfjt mikf gagLwd.

M fhL/ gF tjf kW KjL nghwitfiscsla ,l-ne nkyhik fjaiy rhJsJ.vfhyij f Jakhf ff/ fjij rhjfkhf gagLjKo; MdhY Ja jik W Gfhlhf ,UfhJ vgJcikjh. vD xUt j gzij vthW KjL br-tJ vWrhjdkhf Jakhf KobtLf fj gagL.gndHh whil nrj ghf kW ~bgkh vw ,U fj

iv

tYdf fjij gagL vfhy ffis fFKiwia cUthd. ,U gfilfis Fl jlitf Risaho gntW ff fjffis mtf fzld.

ed bghUshjhu uridf ff fLikmfJ bfhnl nghtjh a Kiwfis VgjFMuh-tjFkhd njit nknkY Tobfhnl nghwJ. fjkW a mogil mikj tKiwfis jfgogagLdh mit Fghf bghUa/ thg kW bjhMa Jiwf RUfkhd/ xik jikila kW wffUfshf mik. nkY ,Kiwf M- br-agL nfhghilMHkhf my Muha cjtnjhlyhk rahd kW gFjmogil fis bgw ttFwd.

2005-2006 f ML Kj mKfgLjgL ,ghl jfgbulh tF tf fj ghll zfvGjgLsJ. xbthU ghlK mogil fU Jtgogoahf fUJ br bgW tz mikfgLsJ. xbthUiyY Vuhskhd vLJfhL fzFf bfhLfgLsd.fUJUfis fiy brhf bghUis khztf eFfWzJ nkY gy fzFfis jhkhfnt vbfhsmbtLJfhLf cj. bfhLfgLs g fzFfkhztfSF nghJkhd gia mF. fzFfis jhfnsf njitahd jdifia tsgjhf mit mik.khztf ,jfij gagLJbghGJ/ clDFl mjjfzFfis Xnuhgoahf nghL ghf ntL vd Unwh.,jf a gFf vf rhj fzLf,Ugjh tf fj khztf mfzFf fSFfghfis (calculators) gagLJkhW mWjgLwhf.jf brhj Kaah gy fzFfis g btbgWkhztf/ a fzFf mogilia czJ mtiw Fmtj w bgUks bgUFtij cWahf ma Ko. bghJnjf ilfis v mf mtfsh ,aY.

,KaF M tH t ela vyh ty ,iwtidnghWnwh. ,jf f r_fdilna tf fjghlfhd Mtij sbjH br- vd enwh.

``mikfhy bghUa jJtfis fLogfja fis neuoahf gagLJ Kiwf fj tYeffuf fwj nrit Msd.pp - M~u kh\kh mk uh#h ,uhk gkehg ,uhkrukh mk uh#h ,uhk gkehg ,uhkrukh mk uh#h ,uhk gkehg ,uhkrukh mk uh#h ,uhk gkehg ,uhkrukh mk uh#h ,uhk gkehg ,uhkru ufh _ unk thr mnjhuh ufh _ unk thr mnjhuh ufh _ unk thr mnjhuh ufh _ unk thr mnjhuh ufh _ unk thr mnjhuh

bghUslfgf

6. tifbfG rkghLftifbfG rkghLftifbfG rkghLftifbfG rkghLftifbfG rkghLf 16.1 tifbfG rkghLfis mikj

tifbfG rkgho tir kW go-tis tiufFLg-rhjhuz tifbfG rkghLfis mikj

6.2 tir xWila tifbfG rkghLftifbfG rkgho -fjf khf-rkgojhdtifbfG rkghLf-tir xW cila rkgojhdtifbfG rkghLfis F Kiw-tir xWila neatifbfG rkghL-bjhifL fhu-kh Fzffisbfhl tir

6.3 kh Fzffis bfhl tir ,uLila neatifbfG rkghLfJiz rkghL kW u rh-w bjhif-bghJ

7. ,ilbrUf kW nenfhL bghUJj 407.1 ,ilbrUf

tiugl Kiw ,ilbrUf fhz-,ilbrUfYfhd,afj Kiwf-lkhd ntWghLf-nfh-lKnehF Nuij jUF Kiw-nfh-lnehF Nuij jUF Kiw-,yuh Nu

7.2 nenfhL bghUJjjw tiugl-W tf bfhif-W tf bfhif_y ,a iy rkghLfis jUj

8. fjf gutffjf gutffjf gutffjf gutffjf gutf 698.1 rkth kh kW fjf rh

jj rkth- kh-jj rkth- kh fjf rhkW fjf gut-F gut rh-bjhl rkth- kh-fjf ml rh-bjhl gut rh

8.2 fza vghj8.3 jj fjf gutf

9. TTTTTbwL cf kW a c-JzjbwL cf kW a c-JzjbwL cf kW a c-JzjbwL cf kW a c-JzjbwL cf kW a c-Jzj 1169.1 TbwLj kW iHf tiff

TbwLj kW TWf-KGikbjhF msit kW TW msit-TbwLj mta-TbwL Kiw l cs gof-TbwLj tiff-TbwL Kiw rhj kW rhuh iHf

9.2 TbwLj gutf,aiy KGik bjhFUJ bgwgL ruhrTbwLj gut-ika viy njw-j msf TbwLjgut-l iH

9.3 kLjkL msit- kL kW ,ilbt kL-KGikbjhF ruhr kW j msfhd eif ,ilbt

9.4 vLnfh nrhjidkWfjf vLnfh kW khW vLnfh-iHf tiff-uhf gF myJ fhL gF kW KaJtkl-KaJt nrhjid

10. gaghL agaghL agaghL agaghL agaghL a 15810.1nea ll

nea ll fz flik-nea ll fzifcUthFj-nea ll gaghLf-y Katiuaiwf-tiugl _y fhz

10.2xLw kW bjhl nghFxLw bghU-jw sfgl-xLwbfG-xLwbfG viyf-bjhl nghF-rhs kh-rhgw kh-,U bjhl nghF nfhLf-

10.3fhyrh bjhl tir gFghfhyrh bjhl tir gFgh- gaf-fhyrh bjhltir TWf-totik-fhy nghid msLj

10.4FblfFL vf tiff-FL vf gaf-FLv mikF j-iwl FL vf-FLvfSfhd nrhjidf-thifju Fbl-thifju Fbliz mikF Kiwf-thif juFbl gaf

10.5a jufLghLkhWghLfSfhd fhuzf-a jufLggho gFkW mj gaf-braghL ju fLghL kW cgbghU ju fLghL-ju fLghL glf

ilfilfilfilfilf 234l ,aiy gutl ,aiy gutl ,aiy gutl ,aiy gutl ,aiy gut 226

7 Kj 10 tiuyhd ghlf cs fzFfis br-afghfis gagLj ntL

vi

1eilKiw uidfis fjto beKiwgLJbghGJ/ mit tif bfGrkghLf totbgW. bghUshjhu/ tfa/ bgha nghw JiwfVgL w ffis FF khf (models) /tifbfG rkghLfshf mikwd. mjifaff bgUghyhdit fyhf kW fodkhf,UF. Mdh ,tiw tif bfG rkgho _ybtgLdh mitg Muh-j f vjhL.vLJfhlhf x cg bghUfSfhd bry khWj bryF neil jkhU ,fit tUtifbfG rkghodh Ffyh.

dxdC

= k C, ,F C vgJ bry kW k xU kh,j thdJ C = C0 ekx , x = 0 vd ,U C = C0 MF.

6.1 tifbfG rkghLfis mikjtifbfG rkghLfis mikjtifbfG rkghLfis mikjtifbfG rkghLfis mikjtifbfG rkghLfis mikj(Formation of differential equations)

xW myJ mjF nkgl rhuh khf/ xU rhj khkW ,t xW myJ mjF nkgltifbfGfis bfhL mikfgL rkghL tifbfG rkghL MF.

tif bfGrkghLf ,U tifgL.

(i) xnubahU rhuh kh/ rhuh khia bghWj rhjkh tifbfG ,itfis jdfnj bfhlrkghL rhjhuz tifbfG rkghlhF.

(ii) xWF nkgl rhuh khf kW rhj khfgF tif bfGf ,tiw bfhL mikrkghoF gF tifbfG rkghL (Partial differentialequation) vW bga.

tifbfG rkghLftifbfG rkghLftifbfG rkghLftifbfG rkghLftifbfG rkghLf 6

2tUtd tifbfG rkghLfSF vLJfhLfshF.

1)2

dxdy

3 dxdy

+ 2y = ex 2) 22

dxyd

5 dxdy

+3y = 0

3)23

2

1

+ dxdy

= k 22

dxyd 4) x

xu

+ y

yu

= 0

5)

2

2

x

u

+ 22

yu

+ 22

zu

= 0 6) 22

x

z

+ 22

yz

= x + y

(1), (2) kW (3) ,itf rhjhuz tifbfGrkghLfshF

(4), (5) kW (6) ,itf gF tifbfGrkghLfshF

,ghl rhjhuz tifbfG rkghLfis gkLnk gongh.

6.1.1 tifbfG rkgho tir kW gotifbfG rkgho tir kW gotifbfG rkgho tir kW gotifbfG rkgho tir kW gotifbfG rkgho tir kW go(Order and Degree of a Differential Equation)xU tif bfG rkgho ,l bgUF tif

bfG f caj tirna mrkgho tir (order)vdgL.

vLJfhlhf/

x23

2

2

dxyd

+ 32

3

3

dxyd

+7 dxdy

4y = 0

vw tifbfG rkghil fU bfhf. 3

3

dxyd

, 2

2

dxyd

kW dxdy

,t tirf Kiwna 3, 2 kW 1 MF.vdnt f caj tir 3. Mifah nkfl tifbfGrkgho tir 3 vd mayh.

xU tifbfG rkgho ,lbgUF fcaj tir bfhl tif bfG gona mrkgho

3go (degree) vdgL. ,jid fhz/ tifbfG rkghocs tifbfGf mLF F/ dkhf,yhkUFkhW cW br-J bfhsntzL.

x23

2

2

dxyd

+32

3

3

dxyd

+7 dxdy

4y = 0 vw tifbfG

rkgho go 2 vdmayh.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 1tU tifbfG rkghLf tirtU tifbfG rkghLf tirtU tifbfG rkghLf tirtU tifbfG rkghLf tirtU tifbfG rkghLf tir

go fhf.go fhf.go fhf.go fhf.go fhf.

(i)3

dxdy

4

dxdy

+ y = 3ex (ii) 32

2dxyd

+ 74

dxdy

= 3sin x

(iii) 22

dyxd

+ a2x = 0 (iv) 2

dxdy

3 33

dxyd

+7 22

dxyd

+4

dxdy

logx= 0

(v)2

1

+ dxdy

= 4x (vi) 32

2

1

+ dxdy

= 2

2

dxyd

(vii) 22

dxyd

dxdy

= 0 (viii) 21 x+ = dxdy

:

tir kW go Kiwna

(i) 1 ; 3 (ii) 2 ; 3 (iii) 2 ; 1 (iv) 3 ; 1(v) 1 ; 2 (vi) 2 ; 3 (vii) 2 ; 2 (viii) 1 ; 1

FFFFF

(v), (vi) kW (vii) ,t go kW tirfisfhgjF K tif bfGf d mLFfis fntL.

6.1.2 tis tiuf FLgtis tiuf FLgtis tiuf FLgtis tiuf FLgtis tiuf FLg (Family of curves)y rkaf/ tistiuf FLgij xnubahU

rkgho _y Ffyh. tistiu FLg

4rkgho c vw ahnjD xU kh ,UF. c -btntW kfSF tistiuFLg btntWtistiufis bgwyh. ,F c -I Jiz myF (parameter)vngh. ,J VnjD xU khahf ,UF.

vLJfhLfvLJfhLfvLJfhLfvLJfhLfvLJfhLf

(i) y = mx vw rkghL Mt brY nenfhLfFLgij FwJ. ,F m xU Jiz myF.

(ii) Mia ikakhfbfhl bghJika tlfFLg x2 + y2 = a2 vw rkghlh FfgLwJ,F a xU Jiz myF.

(iii) xnu js mik nenfhL FLg rkghLy = mx + c vw rkghlh FfgLwJ. ,F mkW c vgd Jiz myFf.

6.1.3 rhjhuz tifbfG rkghLfisrhjhuz tifbfG rkghLfisrhjhuz tifbfG rkghLfisrhjhuz tifbfG rkghLfisrhjhuz tifbfG rkghLfismikj mikj mikj mikj mikj (Formation of Ordinary DifferentialEquations)y = mx + ---------(1) vw rkghil fUJf.

,F m xU kh. xU Jiz myF MF. ,rkghLrkkhd rh-fis bfhl ,iz nfhLfFLgij FF.

(1) x -iabghWJ tifl, dxdy

= m vd ilF. ,J

nenfhL FLg tif bfG rkghil

FwJ. ,nj nghW y = Ae5x vw rkghL dxdy

= 5yvw tifbfG rkghil cUthF.

nkFl rhf xnu xU Jiz myif bfhlFLgfis FF. xbthU FLgF xU tifbfG rkghL cL. ,tif bfG rkghil bgwFLg rkghil x ia bghWJ/ Jiz myifkhahf fU tifL fhz ntL. tifL br-jrkghL Jiz myFf ,UF bghGJ/ FLgtifbfG rkghlhf mik.

5FFFFF(i) ,U Jiz myFf bfhl FLg rkghil

,UKiw tifL br-J Jiz myFfis tifbfG rkghil bgwyh.

(ii) bgwgL tifbfG rkgho tirahdJtistiu FLg rkgho cs khfvifF rkkhf ,UF.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 2y = A cos 5x + B sin 5x vw tistiu FLgvw tistiu FLgvw tistiu FLgvw tistiu FLgvw tistiu FLg

tifbfG rkghiltifbfG rkghiltifbfG rkghiltifbfG rkghiltifbfG rkghil mikf. ,F mikf. ,F mikf. ,F mikf. ,F mikf. ,F A kWkWkWkWkWB Jiz myFfshF.Jiz myFfshF.Jiz myFfshF.Jiz myFfshF.Jiz myFfshF. :

y = A cos 5x + B sin 5x (bfhLfglJ) dx

dy= 5A sin5x + 5B cos 5x

2

2

dxyd

= 25 (A cos 5x) 25 (B sin 5x) = 25y

2

2

dxyd

+ 25y = 0.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 3y = ae3x + bex vw tistiu FLgvw tistiu FLgvw tistiu FLgvw tistiu FLgvw tistiu FLg

tifbfG rkghil fhf. tifbfG rkghil fhf. tifbfG rkghil fhf. tifbfG rkghil fhf. tifbfG rkghil fhf. a kW kW kW kW kW b vgdvgdvgdvgdvgdJiz myFfshF.Jiz myFfshF.Jiz myFfshF.Jiz myFfshF.Jiz myFfshF.

:

y = ae3x + bex ------------(1)

dxdy

= 3ae3x + bex ------------(2)

2

2

dxyd

= 9ae3x + bex ------------(3)

(2) (1) dxdy

y = 2ae3x ------------(4)

6(3) (2) 22

dxyd

dxdy

=6ae3x =3

ydx

dy [(4) ia gagL]

2

2

dxyd

4 dxdy

+ 3y = 0

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 4y = a cos (mx + b), a kW kW kW kW kW b fis VnjDfis VnjDfis VnjDfis VnjDfis VnjD

khfshf bfhl tistiu FLgkhfshf bfhl tistiu FLgkhfshf bfhl tistiu FLgkhfshf bfhl tistiu FLgkhfshf bfhl tistiu FLgtif bfG rkghil fhf.tif bfG rkghil fhf.tif bfG rkghil fhf.tif bfG rkghil fhf.tif bfG rkghil fhf. :

y = a cos (mx + b) ------------(1) dxdy

= ma sin (mx + b)

2

2

dxyd

= m2a cos (mx + b) = m2y [(1)-ia gagL]

2

2

dxyd

+ m2y = 0 vgJ njitahd tifbfG rkghlhF.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 5y = a tan x+b secx vw rkgho csvw rkgho csvw rkgho csvw rkgho csvw rkgho cs

khf khf khf khf khf a kW kW kW kW kW b ,tiw Ftj _y,tiw Ftj _y,tiw Ftj _y,tiw Ftj _y,tiw Ftj _ytifbfG rkghil fhf.tifbfG rkghil fhf.tifbfG rkghil fhf.tifbfG rkghil fhf.tifbfG rkghil fhf.

:

y = a tan x + b sec x,UwK cos x -M bgUf/

y cos x = a sin x + bx -ia bghWJ tifL br-a

y (sin x) + dxdy

cos x = a cos x

y tan x + dxdy

= a -----------(1)(1) ia x ia bghWJ tifL br-a

2

2

dxyd

dxdy tan x y sec2 x = 0

7ggggg 6.11) tUtdt tir kW go fhf.

(i) x2 22

dxyd

3 dxdy

+ y = cos x (ii) 33

dxyd

32

2

2

dxyd

+5 dxdy

= 0

(iii) 22

dxyd

dxdy

= 0 (iv) 21

2

2

1

+

dxyd

= dxdy

(v)31

1

+ dxdy

= 2

2

dxyd (vi)

2

2

1dx

yd+ = x dx

dy

(vii)23

2

2

dxyd

=

2

dxdy (viii) 3 2

2

dxyd

+53

dxdy

3y = ex

(ix) 22

dxyd

= 0 (x) 32

122

+

dxyd

=

31

dxdy

2) tUtdt tifbfG rkghLfis fhf.(i) y = mx (ii) y = cx c + c2(iii) y = mx +

ma

, ,F m xU VnjD khahF

(iv) y = mx + c ,F m kW c vgd VnjD khfshF.3) x , y mRfis bjhiy bjhLnfhLfshf bfhl

mgutisa FLg tifbfG rkghil fhf.

4) Mt brY kW x m J ikafis bfhlx2 + y2 + 2gx = 0 vD tlf FLgij FFtifbfG rkghil fhf.

5) y2 = 4a (x + a) , tifbfG rkghil fhf. ,Fa xU Jiz myF.

6) y = ae2x + be3x vw tistiu FLg tifbfGrkghil fhf. ,F a kW b vgd JizmyFfshF.

7) y = a cos 3x + b sin 3x - tifbfG rkghilfhf. ,F a kW b vgd Jiz myFfshF.

8) y = aebx - tifbfG rkghil fhf. ,F akW b vgd Jiz myFfshF.

89) x2 + y2 = a2 , vw bghJika tlf tifbfGrkghil fhf. ,F a xU Jiz myF.

6.2 tir xWila tifbfG rkghLftir xWila tifbfG rkghLftir xWila tifbfG rkghLftir xWila tifbfG rkghLftir xWila tifbfG rkghLf(First order differential equations)

6.2.1 tifbfG rkgho tifbfG rkgho tifbfG rkgho tifbfG rkgho tifbfG rkgho (Solution offirst order differential equation)bfhLfgl tifbfG rkghoid iw

br-khW mik khfSilnaahd tif bfGfswrh mrkgho MF.

Ys khf vifahdJ/ tifbfGrkgho tirF rkkhf ,U/ midrkgho bghJ (General solution) vngh.

bghJ cs khfSF Fl kfbfhLJ bgwgL F w (Particular solution)vW bga vLJfhlhf/

tifbfG rkghL bghJ w

(i) dxdy

= sec2x y = tan x + c y= tan x - 5 (c xU khahF)

(ii) dxdy

= x2 + 2x y = 33x

+x2+ c y = 33x + x2 + 8

(iii) 22

dxyd

9y = 0 y = Ae3x + Be-3x y = 5e3x7e-3x

6.2.2 fjf khf fjf khf fjf khf fjf khf fjf khf (Variables separable)tir 1 kW go 1 Mf cs tif bfGrkgho

khf jjna ,U fshf mikkhW f jftifU/ mitf fjf khf vdmiHfgL.

khf fgl / tifbfG rkghLf(x) dx + g(y) dy = 0 vw toitbgW. ,F f(x) vgJ x

9ia kL khahfbfhlJ g(y) vgJ y ia kLkhahf bfhlJkhd rhfshf mik.

,j bghJ thdJ f (x) dx + g (y) dy = c MF.(c xU bjhifl khahF)

vLJfhlhf/ x dxdy

y = 0 vw rkghil fUJf.

x dxdy

= y ydy

= x

dx (khfis gjh)

ydy

= xdx + k k / xU bjhifl kh

log y = log x + k.k MdJ Kj tiuyhd kfis bgwyh.k - k Kj tiu miktijnghW log c -

k miktjh bjhifl kh k -F gyhf log cvw khia bghJ khaikg _y Jbgh bgWwJ.

log y log x = log c log (x

y ) = log c(m-J)

x

y = c y = cx

FFFFF(i) tir xWila nea tifbfG rkgho y

,yhkU ,tifbfG rkghL dxdy

= f(x)vw toit bgW ,j y = f (x) dx + c vdmik.

(ii) x ,yhkUif/ dxdy

= g(y) vw toitbgW/thdJ )( yg

dy = dx + c vd mik.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 6xdy + ydx = 0 vw tifbfG rkghilvw tifbfG rkghilvw tifbfG rkghilvw tifbfG rkghilvw tifbfG rkghil

f.f.f.f.f.

10

:

xdy + ydx = 0 [xy M tFf]

ydy

+x

dx= 0. y

dy + x

dx = c1

log y + log x = log c xy = cFFFFF(i) xdy + ydx = 0 d(xy) = 0 xy = c, .

(ii) d( yx ) = 2y

xdyydx

2yxdyydx

= d ( yx ) + c = y

x + c

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 7

f :f :f :f :f : dxdy

= e3x+y

:

dxdy

= e3x ey ye

dy = e3x dx

ye dy =

xe3 dx + c

ey = 33xe

+ c 33xe + e-y = c

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 8 fhf : fhf : fhf : fhf : fhf : (x2 ay) dx = (ax y2)dy

:

bfhLj rkghoUJ

x2dx + y2dy = a(xdy + ydx) x2dx + y2dy = a d(xy)

2x dx +

2y dy= a d (xy) + c 3

3x + 3

3y= a(xy) + c

x3 + y3 = 3axy + c vgJ bghJ MF.

11

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 9 fhf fhf fhf fhf fhf y 2

1)(1 2x+ dy + x 21 y+ dx = 0

:

y 21 x+ dy + x 21 y+ dx = 0 [ 21 x+ 21 y+ M tFf]

21 y

y+

dy + 21 x

x

+dx = 0

+ 21 yy dy +

+ 21 xx dx = c1

21

21

t dt + 21

21

u du = c

(m.J) 21

t + 21

u = c or 21 y+

+ 21 x+ = cFFFFF : ,fzif bjhifliy tU ia

gagL fhzyh. [ f(x)]n f (x) dx = 1)]([ 1

+

+

n

xf n

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 10 fhf : fhf : fhf : fhf : fhf :

(sin x + cos x) dy + (cos x sin x) dx = 0 :

bfhLfgl rkghil tUkhW vGj/

dy + xxxx

cossinsincos

+ dx = 0

dy

+ +

xxxx

cossinsincos dx = c

y + log(sin x + cos x) = cvLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 11

x = 2 vDbghGJ vDbghGJ vDbghGJ vDbghGJ vDbghGJ y = 4pi

vvvvv

x dxdy

+ cos y = 0 I f.I f.I f.I f.I f.

1+y2= t vf2ydy = dt 1+x2= u vf 2xdx= du

12

:

x dy = cos y dx

sec y dy = xdx

+ k k / bjhifl kh

log (sec y + tan y) + log x = log c, ,F k = log c x(sec y + tan y) = c.

x = 2 , y = 4pi

, v

2

(sec 4pi

+ tan

4pi

) = c c =

2

( 2 + 1) = 2 + 2 w thdJ x (sec y + tan y) = 2 + 2 MF.vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 12

x myFf cgfhd ,Wiy brymyFf cgfhd ,Wiy brymyFf cgfhd ,Wiy brymyFf cgfhd ,Wiy brymyFf cgfhd ,Wiy bryrh rh rh rh rh MC = 23+16x 3x2. kW . kW . kW . kW . kW 1 myF cgfhdmyF cgfhdmyF cgfhdmyF cgfhdmyF cgfhdbkhj bry %.40 v/ bkhj bry kWbkhj bry %.40 v/ bkhj bry kWbkhj bry %.40 v/ bkhj bry kWbkhj bry %.40 v/ bkhj bry kWbkhj bry %.40 v/ bkhj bry kWruhr bry rhfis fhf.ruhr bry rhfis fhf.ruhr bry rhfis fhf.ruhr bry rhfis fhf.ruhr bry rhfis fhf.

:

x myFf cg bkhj bry rh C(x) v

dxdC

= MC = 23 + 16x 3x2

dxdC dx = ( 23+16x 3x2)dx+ k C = 23x + 8x2 x3 + k, k xU kh x = 1 v C(x) = 40 (bfhLfglJ)23(1) + 8(1)2 - 13 + k = 40 k = 10

bkhj bry rh C(x) = 23x + 8x2 x3 + 10ruhr bry rh =

xxxx 10823 32 ++

= 23 + 8x x2 + x

10

13

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 13njit be njit be njit be njit be njit be 1 v njit rhv njit rhv njit rhv njit rhv njit rh

bghJ tot fhf.bghJ tot fhf.bghJ tot fhf.bghJ tot fhf.bghJ tot fhf.

:

p iyF nfhugL bghUf vif x vf.

d = xp

dpdx

bfhLfglJ/ x

p

dpdx

= 1 x

dx= p

dp x

dx

= pdp

+ log k

log x = log p + log k, (k xU kh) log x = log kp x = kp p =

k1 x

(m.J) p = cx, ,F c = k1

xU kh

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 14xU jha lif guhkgjfhd bry xU jha lif guhkgjfhd bry xU jha lif guhkgjfhd bry xU jha lif guhkgjfhd bry xU jha lif guhkgjfhd bry c

kW m nrJ itf Toa bghU mskW m nrJ itf Toa bghU mskW m nrJ itf Toa bghU mskW m nrJ itf Toa bghU mskW m nrJ itf Toa bghU ms

x Matiw bjhlgLJ rkghLMatiw bjhlgLJ rkghLMatiw bjhlgLJ rkghLMatiw bjhlgLJ rkghLMatiw bjhlgLJ rkghL dxdC

= ax + b.x = 0 vD nghJ vD nghJ vD nghJ vD nghJ vD nghJ C = C0 v v v v v C -I -I -I -I -I x - rhghf- rhghf- rhghf- rhghf- rhghffhf.fhf.fhf.fhf.fhf.

:

dxdC

= ax + b dC = (ax + b) dx

Cd = + )b(ax dx + k,

C = 22ax +bx + k, (k xU kh) -------------(1)

x = 0 v C = C0 (1) C0 = 2a (0) + b(0) + k

k = C0vdnt bry rh C = 2

ax2 + bx + C0

14

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 15xU tistiu vj xU Y mjxU tistiu vj xU Y mjxU tistiu vj xU Y mjxU tistiu vj xU Y mjxU tistiu vj xU Y mj

rh- m rh- m rh- m rh- m rh- m y Mabjhiy ,UklMabjhiy ,UklMabjhiy ,UklMabjhiy ,UklMabjhiy ,Ukljiy MF. tistiu jiy MF. tistiu jiy MF. tistiu jiy MF. tistiu jiy MF. tistiu (4, 3) t brYwJt brYwJt brYwJt brYwJt brYwJv mj rkghil fhf.v mj rkghil fhf.v mj rkghil fhf.v mj rkghil fhf.v mj rkghil fhf.

:

xU lJ tistiu rh- vgJ m

bjhLnfho rh-thF.

dxdy

= y21

2ydy = dx

y2 dy = dx + c y2 = x + ctistiu (4, 3) vw tna brtjh

9 = 4 + c c = 5 tistiu rkghL y2 = x + 5

ggggg 6.2

1) f (i) dxdy

+ 2

2

11

x

y

= 0 (ii) dxdy

= 2

2

11

x

y+

+

(iii) dxdy

= 12

+x

y (iv) x 21 y+ + y 21 x+ dx

dy= 0

2) f (i) dxdy

= e2x-y + x3 ey (ii) (1ex) sec2 y dy + 3ex tan y dx = 0

3) f (i) dxdy

= 2xy + 2ax (ii) x(y2 + 1) dx + y(x2 + 1) dy = 0

(iii) (x2 yx2) dxdy

+ y2 + xy2 = 0

4) f (i) xdy + ydx + 4 221 yx dx =0 (ii) ydxxdy+3x2y2ex3dx = 0

5) f (i) dxdy

=

2254

2

2

+

++

xx

yy (ii) dxdy

+11

2

2

++

++

xx

yy = 0

15

6) P(x, y) vw lJ xU tistiu rh- 3x2 + 2MF. tistiu (1, -1) tbrYbk mjrkghil fhf.

7) (x , y) vw xU mj rh- m xMabjhiyF ne jrk csJ. tistiu (0, 0)kW (1, 1) vD f t brYwJ v mjrkghil fhf.

8) x = 21 vD nghJ y = 2 v sin-1x dy +

21 xy

dx = 0 If.

9) njit be - n v njit rkgho bghJ totfhf.

10) njit be 21

v njit rkgho bghJ

tot fhf.

11) x myFf cgfhd ,Wiy bry rhMC = e3x + 7. cg VJ ,yhjnghJ bkhj bry,iy vdbfhL/ bkhj bry kW ruhr bry

rhfis fhf.

6.2.3 rkgojhd tifbfG rkghLfrkgojhd tifbfG rkghLfrkgojhd tifbfG rkghLfrkgojhd tifbfG rkghLfrkgojhd tifbfG rkghLf(Homogeneous differential equations)f(x, y) kW g(x, y) vgd xbthW xnu gos

rkgojhd rhfbs dxdy

= ),(),(

yxgyxf

vgJ x, y , xU

rkgojhd tifbfG rkghL MF.

dxdy

= 22 yxxy+

, dxdy

= xy

yx2

22 +, dx

dy = 33

2

yxyx

+

kW dxdy

=

x

yyx + 22

vgd tir xW cila rkgojhd tifbfGrkghLfSF y vLJ fhLfshF.

16

6.2.4 tir xW cila rkgojhd tifbfGtir xW cila rkgojhd tifbfGtir xW cila rkgojhd tifbfGtir xW cila rkgojhd tifbfGtir xW cila rkgojhd tifbfGrkghLfis F Kiw rkghLfis F Kiw rkghLfis F Kiw rkghLfis F Kiw rkghLfis F Kiw (Solving first orderhomogeneous differential equations)y = vx v dx

dy = v + x dx

dv MF. vdnt tir xW

cila rkgojhd tifbfG rkghlhdJ khfisf Toa rkgho tot bgW. bjhifLjYF

wF v I x

y vW kh it fhzyh.

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 16 tU tifbfG rkghil f. tU tifbfG rkghil f. tU tifbfG rkghil f. tU tifbfG rkghil f. tU tifbfG rkghil f.(x2 + y2)dx = 2xydy

:

bfhLfgl tifbfG rkghil

dxdy

= xy

yx2

22 + vW vGjyh. ------------- (1)

,J xU rkgojhd tifbfG rkghL MF.

y = vx vf dxdy

= v + x dxdv

------------- (2)(1) I (2) , ul

v + x dxdv

= )(2222

vxxxvx +

= vv

21 2+

x dxdv

= vv

21 2+

v x dxdv

= vv

21 2

khfis gjh/

212

v

v

dv= x

dx

212v

v =

xdx

+ c1

log (1 v2) = log x + log c [

)()(

xfxf

dx = log f(x)] log (1 v2) + log x = log c (1 v2) x = cv I

x

y vd khdh

2

2

1x

yx = c myJ x2 y2 = cx vd mik.

17

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 17fffff : (x3 + y3)dx = (x2y + xy2) dy

:

bfhLfgl rkghil tUkhW vGjyh.

dxdy

= 22

33

xyyxyx

+

+-------------- (1)

y = vx vf. dxdy

= v + x dxdv

v + x dxdv

= 2

31vv

v

++

x dxdv

= 2

31vv

v

++

v = )1(1 2

+

vvv

= )1()1)(1(

++

vv

vv

vv

1 dv =

x1

dx + c

vv

1 dv =

x1

dx + c or

v

v

11)1(

dv = x1 dx + c

+v1)1(1 dv = x

1 dx + c

v + log (1 v) = log x + c v I

x

y vd khdh

x

y+ log (x y) = c MF.

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 18

fffff : x dxdy

= y 22 yx +

:

dxdy

=

x

yxy 22 +------------(1)

y = vx vf dxdy

= v + x dxdv

(1) v + x dxdv

=

xxvxvx 222 +

= v 21 v+

x dxdv

= 21 v+ = 21 vdv+

= x

dx

18

+ 21 vdv

= xdx

+ c1

log (v + 21 v+ )= log x + log c log x (v + 21 v+ )= log c

myJ x (v + 21 v+ ) = c

(m-J) x

++ 2

2

1x

yx

y= c y + 22 yx + = c

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 19fffff (x +y) dy + (x y)dx = 0

:

dxdy

=

+

yxyx

------------ (1)

y = vx vf. dxdy

= v + x dxdv

vdnt v + x dxdv

= vxxvxx

+

v + x

dxdv

= vv

+

11

(m-J) x dxdv

=

+

+ v

vv

11

x dxdv

=v

vvv

+++

1)1( 2

211

v

v

++ dv =

x1

dx

+ 21 vdv dv + + 21

221

v

v dv = x1

dx + c

tan-1v + 21 log (1 + v2) = log x + c

tan-1

x

y + 2

1 log

+2

22

x

yx = logx + c

tan-1

x

y + 2

1 log (x2 + y2) 21 logx2 = logx + c

tan-1

x

y + 2

1 log (x2 + y2) = c

19

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 20,yhg,yhg,yhg,yhg,yhg p kW nfhugL njit mskW nfhugL njit mskW nfhugL njit mskW nfhugL njit mskW nfhugL njit ms

x MaitMaitMaitMaitMait dxdp

= 2

33

32

xpxp

vw tifbfG rkvw tifbfG rkvw tifbfG rkvw tifbfG rkvw tifbfG rk

ghil iw br-wd. ghil iw br-wd. ghil iw br-wd. ghil iw br-wd. ghil iw br-wd. x = 10 vD nghJ vD nghJ vD nghJ vD nghJ vD nghJ p = 20v/ ,yhg kW nfhugL nv/ ,yhg kW nfhugL nv/ ,yhg kW nfhugL nv/ ,yhg kW nfhugL nv/ ,yhg kW nfhugL njit MatWjit MatWjit MatWjit MatWjit MatWilnas bjhlig fhf.ilnas bjhlig fhf.ilnas bjhlig fhf.ilnas bjhlig fhf.ilnas bjhlig fhf.

:

dxdp

= 2

33

32

xpxp

-------------(1)vgJ x, p , rkgojhd tifbfG rkghL MF.

p = vx v dxdp

= v + x dxdv

.

(1) v + x dxdv

= 2

3

312

v

v x dx

dv= 2

3

312

v

v v

x dxdv

=

+2

3

31

v

v

3

2

13

v

v

+dv=

xdx

+ 32

13

v

v dv = xdx

= k

log (1 + v3) = log x + log k , ,F k xU khlog (1 + v3) = log

xk

1 + v3 =xk

v I x

p vd khdh/

x3 + p3 = kx2

Mdh x = 10 v p = 20 vd bfhLfgLsJ. (10)3 + (20)3 = k(10)2 k = 90 x3+p3 = 90x2(m-J) p3 = x2 (90 x) vgJ njitahd bjhlghF..vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 21

bghUf nfhUj ms bghUf nfhUj ms bghUf nfhUj ms bghUf nfhUj ms bghUf nfhUj ms q mfFbghGJ/mfFbghGJ/mfFbghGJ/mfFbghGJ/mfFbghGJ/nfhUj kW mitfis ,U itgjFkhdnfhUj kW mitfis ,U itgjFkhdnfhUj kW mitfis ,U itgjFkhdnfhUj kW mitfis ,U itgjFkhdnfhUj kW mitfis ,U itgjFkhd

20

bry bry bry bry bry C - mfF j mfF j mfF j mfF j mfF j dqdC

= 2

2 2CCq

q+ vDvDvDvDvD

tifbfG rkghodh jugLsJ. tifbfG rkghodh jugLsJ. tifbfG rkghodh jugLsJ. tifbfG rkghodh jugLsJ. tifbfG rkghodh jugLsJ. C kWkWkWkWkWq F ,ilna cs bjhlig F ,ilna cs bjhlig F ,ilna cs bjhlig F ,ilna cs bjhlig F ,ilna cs bjhlig C = 1 kW kW kW kW kW q = 1vD iy fhf.vD iy fhf.vD iy fhf.vD iy fhf.vD iy fhf.

:

C kW q cs rkgojhd rkghL

dqdC

= 2

2 C2Cq

q+------------(1)

C = vq vf dqdC

= v + q dqdv

(1) v + q dqdv

= 2

222 2q

vqqv + = v2 + 2v

q dqdv

= v2 + v = v (v + 1) )1( +vvdv

= qdq

++)1(

)1(vv

vv dv = q

dq + k , k xU kh.

vdv

+1vdv

= qdq

+ log k,

log v log (v + 1) = log q + log k log 1+v

v = log qk myJ 1+v

v= kq

v = qC

vD nghJ, C = kq(C + q).C = 1 kW q = 1 vD nghJ

C = kq(C + q) k = 21

C = 2)C( qq +

vgJ C kW q F ,ilyhdbjhlghF.

21

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 22bkhj cg bry bkhj cg bry bkhj cg bry bkhj cg bry bkhj cg bry y kW cgkW cgkW cgkW cgkW cg

ms ms ms ms ms x Mait Mait Mait Mait Mait (6x2 + 2y2) dx (x2 + 4xy) dy = 0 vwvwvwvwvwrkgho thyhf ,W iy cg brylrkgho thyhf ,W iy cg brylrkgho thyhf ,W iy cg brylrkgho thyhf ,W iy cg brylrkgho thyhf ,W iy cg brylbjhlgLjgLsd. bjhlgLjgLsd. bjhlgLjgLsd. bjhlgLjgLsd. bjhlgLjgLsd. x =1 vD bghGJ vD bghGJ vD bghGJ vD bghGJ vD bghGJ y = 2v/ bkhj bryF cgF ,ilnav/ bkhj bryF cgF ,ilnav/ bkhj bryF cgF ,ilnav/ bkhj bryF cgF ,ilnav/ bkhj bryF cgF ,ilnas bjhlid fhf.s bjhlid fhf.s bjhlid fhf.s bjhlid fhf.s bjhlid fhf.

:

bfhLfgLsJ (6x2 + 2y2) dx = (x2 + 4xy) dy

dxdy

= xyxyx

426

2

22

+

+------------(1)

,J x kW y - cs rkgojhd rkghL MF.

y = vx vf. dxdy

= v +x dxdv

(1) v + x dxdv

=xyxyx

426

2

22

+

+ 226

)41(vv

dvv

+ dv =

x1

dx

22641

vv

v dv = x1 dx + k, k xU kh

log(6v2v2) = log x + log k = log kx 226

1vv

= kx

x = c(6x2 xy 2y2) ,F c = k1

kW v = x

y.

x = 1 kW y = 2 v, 1 = c(6 2 8) c = 41

4x = (2y2 + xy 6x2)ggggg 6.3

1) tU tifbfG rkghLfis f.(i) dx

dy =

x

y 2

2

x

y (ii) 2 dxdy

= x

y 2

2

x

y

(iii) dxdy

= xyxyxy

32

2

2

(iv) x(y x) dxdy

= y 2

22

(v) dxdy

= xyxxyy

22

2

2

(vi) dxdy

= 22 yxxy

(vii) (x + y)2 dx = 2x2 dy (viii) x dxdy

= y + 22 yx +

2) bghUf nfhUj ms q mfFbghGJ/ nfhUjkW mitfis ,U itgjFkhd bry C -

mfF j dqdC

= qq

C2C 22 +

vD tifbfG

rkghodh jugLsJ. C kW q F ,ilna csbjhlig C = 4 kW q = 2 vD iy fhf.

3) bkhj cg bry y kW cg ms x Mait

dxdy

= xy

yx 2224 vw tifbfG rkgho thyhf

,W iy cg bryl bjhlgLjgLsd.

x = 2 vD bghGJ y = 4 v/ bkhj bry rh ahJ?6.2.5 tir xWila nea tifbfGtir xWila nea tifbfGtir xWila nea tifbfGtir xWila nea tifbfGtir xWila nea tifbfG

rkghL rkghL rkghL rkghL rkghL (First order linear differential equation)tir xW cs tif bfG rkgho cs rhj

kh kW mj tif bfGf go 1 kW ,uobgUf gy ,yhkY ,UFkh mrkghL tirxWila nea tifbfG rkghL vdgL.

tir 1 cs nea tifbfG rkgho tot

dxdy

+ Py = Q vd mik. ,F P kW Q vgd x kLnkcs rhfshF.

vLJfhlhf/

(i) dxdy

+3y = x3 ; ,F P = 3, Q = x3

(ii) dxdy

+ y tan x = cos x, P = tan x, Q = cos x

(iii) dxdy

x 3y = xex, P = x3

, Q = ex

(iv) (1 + x2) dxdy

+ xy = (1+x2)3, P = 21 xx

+, Q = (1 + x2)2

vgd tir 1 cs nea tif bfG rkghLfshF.

23

6.2.6 bjhifL fhu bjhifL fhu bjhifL fhu bjhifL fhu bjhifL fhu (I.F)bfhLfgl tifbfG rkghL/ neilahf

bjhifL fhzjf tif ,yhk ,Ufyh. Mdh,ij xU rh _y bgUFtjh bjhifLfhzjfjhf khwyh. ,jifa rhF bjhifLfhu (integrating factor (I.F)) vW bga. Mifah/tifbfG rkghil neilahf bjhifL fhzjftif khWtjF cj rh bjhifL fhuahF.

dxdy

+ Py = Q, -----------(1) vD rkghoF dxe P

vgJ bjhifL fhu vd Wnth. ,F P kW Qvgd x , rhf.

dxd )( P dxye = dx

dy dxe P + y dx

d ( dxe P )

= dxdy dxe P

+ y dx

eP

dxd dxP

= dxdy dxe P

+ y dx

eP

P = ( dxdy

+Py) dxe P .

(1) ia dxe P M bgUf( dx

dy + Py) dxe P = Q dxe P

dxd )( P dxye = Q dxe P

bjhifL br-a/

y dx

eP

= dxe

P Q

dx + c -------------(2)Mifah

dxe

p vgJ bjhifL fhuahF.

FFFFF

(i) elogf(x) = f(x) when f(x) > 0(ii) dx

dy + Py = Q - Q = 0 v bghJ y (I.F) = c,

,F c xU kh

24

(iii) dydx

+ Px = Q vw rkgho P kW Q vgdy - rhfsh ,rkgho I.F

dye

P MF.

kW thdJ x (I.F) = Q (I.F) dy + c vd mik.vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 23

(1 x2) dxdy

xy = 1 vw rkghil f.vw rkghil f.vw rkghil f.vw rkghil f.vw rkghil f.

:

(1-x2) dxdy

xy = 1

dxdy

21 xx

y = 211x

,J dxdy

+ Py = Q, vw to csJ

,F P = 21 xx

; Q =

211x

I.F = dx

eP

=

dxx

x

e21

= 21 x

bghJ:

y (I.F) = Q (I.F)dx + cy 21 x =

211x

21 x

dx + c

=

21 xdx

+ c

y 21 x = sin-1x + c


fffff dxdy

+ay = ex (,F ,F ,F ,F ,F a 1) :

,F P = a ; Q = ex I.F =

dxe

P = eax

25

bghJ

y (I.F) = Q (I.F)dx + c y eax =

xe eax dx + c = + xae )1( dx + c

y eax = 1)1(

+

+

ae xa

+ c


fffff cos x dxdy

+ y sin x = 1

:

bfhLfgl rkghil tU Kiw vGj

dxdy

+ yxx

cossin

= xcos

1 dx

dy+ y tan x = sec x

,F P = tan x ; Q = sec x I.F =

dxxe

tan = elog secx = sec x

bghJ

y (I.F) = Q (I. F) dx + cy sec x = x

2sec dx + c

y sec x = tan x + c


xU t/ Kjo cldo khW jK/xU t/ Kjo cldo khW jK/xU t/ Kjo cldo khW jK/xU t/ Kjo cldo khW jK/xU t/ Kjo cldo khW jK/mKj XuhL to rk msmKj XuhL to rk msmKj XuhL to rk msmKj XuhL to rk msmKj XuhL to rk ms,UFkhW fzL toawJ. tUlF,UFkhW fzL toawJ. tUlF,UFkhW fzL toawJ. tUlF,UFkhW fzL toawJ. tUlF,UFkhW fzL toawJ. tUlF6.5% bjhl TL j % .bjhl TL j % .bjhl TL j % .bjhl TL j % .bjhl TL j % .50,000 - ij-ij-ij-ij-ijmt xUt KjL br - j h mt xUt KjL br - j h mt xUt KjL br - j h mt xUt KjL br - j h mt xUt KjL br - j h 10tUlfSF d mt bgW KtUlfSF d mt bgW KtUlfSF d mt bgW KtUlfSF d mt bgW KtUlfSF d mt bgW Kbjhifia fzLf.bjhifia fzLf.bjhifia fzLf.bjhifia fzLf.bjhifia fzLf. (e.65 =1.9155)

26

:

t vw fhy P(t) vgJ fz ,UF bjhifvd bfhf. gz tsia FF tifbfGrkghlhdJ

dtdP

= 1005.6 P = 0.065P P

Pd= )065.0( dt + c

logeP = 0.065t + c P = e0.065t ec

P = c1 e0.065t

-------------(1)t = 0, P = 50000 v

(1) 50000 = c1 e0 or c1 = 50000 P = 50000 e0.065t

At t = 10 v P = 50000 e 0.065 x 10 = 50000 e 0.65

= 50000 x (1.9155) = %.95,775vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 27

fffff : dxdy

+ y cos x = 21 sin 2x

:

,F P = cos x ; Q = 21

sin 2x

P dx = xcos dx = sin x

I.F = dx

eP

= e sin x

bghJ

y (I.F) = Q (I.F) dx + c= 2

1sin 2x. esin x dx + c

= xsin cos x. esin x dx + c

= t et dt + c = et (t 1) + c= esin x (sin x 1) + c

sin x= t, vf cos x dx = dt

27

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 28jah Wtd x cgfuzfisjah Wtd x cgfuzfisjah Wtd x cgfuzfisjah Wtd x cgfuzfisjah Wtd x cgfuzfis

,af guhkf MF bry ,af guhkf MF bry ,af guhkf MF bry ,af guhkf MF bry ,af guhkf MF bry C kWkWkWkWkWmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbt

fhy fhy fhy fhy fhy m Matiw Matiw Matiw Matiw Matiw m2 dmdC

+ 2mC = 2, m = 2 vvvvv C = 4vD rkghodh Fjh/ vD rkghodh Fjh/ vD rkghodh Fjh/ vD rkghodh Fjh/ vD rkghodh Fjh/ C kWkWkWkWkW m fSfSfSfSfSilnaahd bjhlig fhf.ilnaahd bjhlig fhf.ilnaahd bjhlig fhf.ilnaahd bjhlig fhf.ilnaahd bjhlig fhf. :

m2 dmdC

+ 2mC = 2 dmdC +

mC2

= 22

m

,J tir 1 cs nea tif bfG rkghlhF.

dxdy

+ Py = Q, ,F P =m2

; Q = 22m

I.F =

dmeP

= dme m

2

= elog m2 = m2

bghJ :

C (I.F) = Q (I.F) dm + k , k xU kh Cm2 = 2

2m

m2 dm + k

Cm2 = 2m + kC = 4 kW m = 2 v

16 = 4 + k k = 12C kW m -fhd bjhlCm2 = 2m + 12 = 2(m + 6) MF.

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 29jah Wtd x cgfuzfisjah Wtd x cgfuzfisjah Wtd x cgfuzfisjah Wtd x cgfuzfisjah Wtd x cgfuzfis

,af guhkf MF bry ,af guhkf MF bry ,af guhkf MF bry ,af guhkf MF bry ,af guhkf MF bry C kWkWkWkWkWmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbtmLjLj ,U gGJ ghjYFa ,ilbt

fhy fhy fhy fhy fhy x Matiw Matiw Matiw Matiw Matiw x2

dxdC

10xC = 10 vwvwvwvwvwrkghodh Ffgorkghodh Ffgorkghodh Ffgorkghodh Ffgorkghodh Ffgo , x = x0 vvvvv C = C0vdbfhL vdbfhL vdbfhL vdbfhL vdbfhL C -I I I I I x - rhghf fhf. rhghf fhf. rhghf fhf. rhghf fhf. rhghf fhf.

28

:

x2 dxdC

10xC = 10 dxdC

xC10

= 210x

,J tir 1 cs nea tifbfG rkghlhF

P = x

10

kW Q = 210x

P dx = x10 dx = 10 log x = log

101

x

I.F = dx

eP

=

101log

xe = 101

xbghJ :

C(I.F) =

Q

(I.F) dx + k, k xU kh

10Cx

=

210x

101

x

dx + k or 10Cx

=

1110

111

x+ k

vdnt C = C0 kW x = x0 v

100

0Cx

= 1110

110

1x

+ k k = 100

Cx

11011

10x

10Cx

=

1110

111

x +

11

0100 11

10Cxx

10Cx

100

Cx

= 1110

11

011

11xx

ggggg 6.41) tU tifbfG rkghLfis f.

(i) dxdy

+ y cot x = cosec x

(ii) dxdy

sin 2x = y cot x

(iii) dxdy

+ y cot x = sin 2x

29

(iv) dxdy

+ y cot x = 4x cosec x, (x = 2pi

v y = 0)

(v)

dxdy

3y cot x = sin 2x , (x = 2pi

v y = 2)

(vi) x

dxdy

3y = x2

(vii) dxdy

+ 212

x

xy+

= 22 )1(1x+

, (x = 1 v y = 0)

(viii) dxdy

y tan x = ex sec x

(ix) log x dxdy

+x

y = sin 2x

2) bjhl TL to j 12% bfhl W nrl xUt 5 tUlfSF xU Fl bjhifiaKjL br-a lLwh. 5 tUlfSF wF %. 25,000ilgjF ,l mt vts gz KjL br-a

ntL vgij fhf. (e0.6 = 0.5488)3) jah Wtd x cgfuzfis ,af

guhkf MF bry C kW mLjLj ,U gGJghjYFa ,ilbt fhy x Matiw ,izF

rkghL x2 dxdC

(b1)Cx = ba MF. , a, b Madkhf. kW x = x0 v C = C0 MF. C kW x,t bjhlfhd rkghil fhf.

4) bghU ms q khWbghGJ/ nfhUj kW ,Uitj Matfhd bry C - khWjiy dq

dC= a q

C,

(a xU kh) vD rkghodh Fjh/ q = q0 vDbghGJ C = C0 vd bfhL C -ia q - rhghf fUJf.

6.3 kh Fzffis bfhl tirkh Fzffis bfhl tirkh Fzffis bfhl tirkh Fzffis bfhl tirkh Fzffis bfhl tir,uLila nea tifbfG rkghLf,uLila nea tifbfG rkghLf,uLila nea tifbfG rkghLf,uLila nea tifbfG rkghLf,uLila nea tifbfG rkghLf

(Second order linear differential equationswith constant co-efficients)

tir 2 cs/ kh Fzffis bfhl/ neatif bfG rkgho bghJ tot:

30

a 2

2

dxyd

+ b dxdy

+ cy = f(x).,gF (i) f(x) = 0 (ii) f(x) = kex vd mik

tifbfG rkghLfis kLnk fUJnth.

vLJfhlhf/

(i) 3 22

dxyd

5 dxdy

+ 6y = 0 (or) 3y`` 5y` + 6y = 0

(ii) 22

dxyd

4 dxdy

+ 3y = e5x (or) (D2 4D + 3)y = e5x

(iii) 22

dxyd

+ dxdy

y = 7 (or) (D2 + D 1)y = 7vgd tir ,uLila nea rkghLfshF.

6.3.1 Jiz rkghL kW u rhJiz rkghL kW u rhJiz rkghL kW u rhJiz rkghL kW u rhJiz rkghL kW u rh(Auxiliary equation and Complementary function)

a 2

2

dxyd

+ b dxdy

+ cy = f (x ) , vw tifbfGrkghoF am2 + bm + c = 0 vgij/ Jiz rkghL(auxiliary equation) vnwh. ,J m - ,Ugo rkghlhF.,rkgho fshd m1 kW m2 jikF Vgu rhf (complementary function) tU jmik.

f jikf jikf jikf jikf jik urh urh urh urh urh

(i) bk- kW btntW Aem1x + Bem2x

(m1 m2)(ii) bk- kW rkkhdJ (Ax + B) emx

(m1 = m2=m v)(iii) fy vf ( + i) ex(Acos x + Bsin x)

(A kW B vgd VnjD khfshF.

31

6.3.2 w bjhif w bjhif w bjhif w bjhif w bjhif (P.I)(aD2 + bD + c)y = ex vgij fUJff(D) = aD2 + bD + c vf.

tiftiftiftiftif 1 : f() 0 v Jiz rkghL f(m) = 0 -F xU _y my.

KiwKiwKiwKiwKiw : P.I = )D(1

f ex = )(1f e

x.

tiftiftiftiftif 2 : f() = 0 v/ f(m) = 0 , _y MF.,iy/

(i) Jiz rkgho f m1 kW m2 vf. nkY = m1 vd bfhf. f(m) = a(m m1) (m m2) = a(m ) (m m2)

KiwKiwKiwKiwKiw : P.I = ))(-D(1

2mDa ex

= )(1

2ma xex

(ii) Jiz rkghL MdJ ,uL rkkhd fisbgU (m.J) m1 = m2 = v f(m) = a ( m )2

KiwKiwKiwKiwKiw : P.I. = 2)D(1

a ex

=

! 21 2xa

ex

6.3.3 bghJ bghJ bghJ bghJ bghJ tir 2 cs nea tifbfG rkgho bghJ

y = u rh (C.F) + w bjhif (P.I) MF.vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 30

fffff : 3

2

2

dxyd

5 dxdy

+ 2y = 0 :

Jiz rkghL 3m2 5m + 2 = 0 (3m 2) (m 1) = 0

f m1 = 32

kW m2 = 1 (bk- kW btntW)

32

u rh

C.F = Ax

e 32

+ Bex

bghJ

y = Ax

e 32

+ Bex .

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 31fffff : (16D2 24D + 9) y = 0

:

Jiz rkghL 16m2 24m + 9 = 0 (4m -3)2 = 0 m = 4

3, 4

3 (f rk)

C.F = (Ax + B) xe 43

bghJ y = (Ax + B) xe 43

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 32fffff : (D2 6D + 25) y = 0

:

Jiz rkghL m2 6m + 25 = 0

m = a

acbb2

42

= 2100366

= 286 i

= 3 + 4i

f + i toYs fy vf. ,F = 3kW = 4 MF.

C.F = ex (A cos x + B sin x) u rh = e3x (A cos 4x + B sin 4x)bghJ :

y = e3x (A cos 4x + B sin 4x)

33


fffff :

2

2

dxyd

5 dxdy

+ 6y = e5x

:

Jiz rkghL m2 - 5m + 6 = 0 m = 3 , 2 urh C. F = Ae3x + Be2x

w bjhif P. I = 6D5D1

2 + e5x = 6

1e5x

bghJ

y = C.F + P. I

y = Ae3x + Be2x + 65xe


fffff : 2dxyd 2

+ 4 dxdy

+ 4y = 2e-3x

:

Jiz rkghL m2 + 4m + 4 = 0 m = 2, 2 u rh C. F = (Ax + B)e2x

w bjhif P. I = 4D4D

12 ++

2e3x

=

4)3(4)3(1

2 ++ 2e3x = 2e3x

bghJ

y = C.F + P. Iy = (Ax + B) e2x + 2e3x


fffff : 2

2

dxyd

2 dxdy

+ 4y = 5 + 3e-x

:

Jiz rkghL m2 2m + 4 = 0

34

m =2

1642 =

2322 i

= 1 + i 3

C.F = ex (A cos 3 x + B sin 3 x)w bjhif P. I1 = 4D2D

12 +

5 e0x = 41 5 e0x = 4

5

w bjhif P.I2 = 4D2D1

2 +3 ex

=

4)1(2)1(1

2 + 3e-x = 7

3 xe

bghJ

y = C.F + P. I1 + P.I2y = ex (A cos 3 x + B sin 3 x) + 4

5 + 7

3 e-x

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 36 f f f f f : (4D2 - 8D+ 3)y = xe 2

1

:

Jiz rkghL 4m2 - 8m + 3 = 0

m1 = 23

, m2 = 21

u rh C.F = Ax

e 23

+ Bx

e 21

w bjhif P. I = 3D84D1

2 +

xe 2

1

=

)21

-D)(23

-4(D1 x

e 21

= )D)(4(1

21

23

21

xe 2

1 = 4

x xe 2

1

bghJ

y = C.F. + P. Iy = A

xe 2

3

+ Bx

e 21

4x 2

x

e

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 37fffff : (D2 + 10D + 25)y = 2

5 + e-5x

35

:

Jiz rkghL m2 + 10m + 25 = 0 (m + 5)2 = 0 m = 5, 5

u rh C.F = (Ax + B) e5x

w bjhif P. I1 = 25D10D1

2 ++ 25

e0x

= 251

25

= 101

w bjhif P.I2 = 25D10D1

2 ++e5x = 2)5D(

1+ e

5x

= 22x

(e5x) bghJ

y = C.F + P. I1 + P.I2y = (Ax + B) e5x + 10

1 + 2

2x e5x

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 38

Qd = 42 4p 4 dtdp

+ 2

2

dtpd

kW kW kW kW kW Qs = -6 + 8p

vgd Kiwna xU bghU njit msvgd Kiwna xU bghU njit msvgd Kiwna xU bghU njit msvgd Kiwna xU bghU njit msvgd Kiwna xU bghU njit mskW m ms Madtiw FwJ.kW m ms Madtiw FwJ.kW m ms Madtiw FwJ.kW m ms Madtiw FwJ.kW m ms Madtiw FwJ.(,F (,F (,F (,F (,F p iyia FwJ) rij gkhwiyia FwJ) rij gkhwiyia FwJ) rij gkhwiyia FwJ) rij gkhwiyia FwJ) rij gkhwrkiy iyia rkiy iyia rkiy iyia rkiy iyia rkiy iyia (equilibrium price) fhf. fhf. fhf. fhf. fhf. :

rk iy iy/ Qd = Qs Mf ,UF. 42 4p 4 dt

dp + 2

2

dtpd

= 6 + 8p

48 12p 4 dtdp

+ 2

2

dtpd

= 0

2

2

dtpd

4 dtdp

12p = -48

Jiz rkghL m2 4m 12 = 0

36

m = 6 , 2 C.F. = Ae6t + Be-2t

P. I = 12D4D1

2

(48) e0t = 121

(48) = 4bghJ

p = C.F. + P. I p = Ae6t + Be-2t + 4ggggg 6.5

1) tU tifbfG rkghLfis f.(i)

2

2

dxyd

- 10 dxdy

+ 24y = 0 (ii) 22

dxyd

+ dxdy

= 0

(iii) 22

dxyd

+ 4y = 0 (iv) 22

dxyd

+ 4 dxdy

+ 4y = 0

2) f :(i) (3D2 + 7D - 6)y = 0 (ii) (4D2 12D + 9)y = 0(iii) (3D2 D + 1)y = 0

3) f :(i) (D2 13D + 12) y = e2x + 5ex(ii) (D2 5D + 6) y = ex + 3e2x(iii) (D2 14D + 49)y = 3 + e7x

(iv) (15D2 2D 1)y = 3xe

4) Qd = 305P + 2 dtdP

+ 22P

dtd

kW Qs = 6 + 3P vgd Kiwna

xU bghU njit ms kW m ms

Madtiw Fwd. ,F p iyia FwJ.rij gkh rkiy iyia fhf.

ggggg 6.6Vila ilia bj br-f.Vila ilia bj br-f.Vila ilia bj br-f.Vila ilia bj br-f.Vila ilia bj br-f.

1) M t brY nenfhLf tifbfG rkghL(a) x dx

dy = y (b) dx

dy = y

x (c) dxdy

= 0 (d)x dxdy

= y1

37

2) 22

dxyd

-6 dxdy

= 0 vw tifbfG rkgho go kWtir Kiwna

(a) 2 kW 1 (b) 1 kW 2 (c) 2 kW 2 (d) 1 kW 1

3)2

dxdy

3 33

dxyd

+ 7 22

dxyd

+ dxdy

= x + log x vw tifbfGrkgho tir kW go Kiwna

(a) 1 kW 3 (b) 3 kW 1 (c) 2 kW 3 (d) 3 kW 2

4)32

2

1

+ dxdy

= 2

2

dxyd

vw rkgho tir kW go

Kiwna

(a) 3 kW 2 (b) 2 kW 3 (c) 3 kW 3 (d)2 kW 25) x dy + y dx = 0

(a) x + y = c (b) x2 + y2 = c (c) xy = c (d) y = cx6) x dx + y dy = 0

(a) x2 + y2 = c (b) yx

= c (c) x2 y2 = c (d) xy = c

7) dxdy

= ex y

(a) ey ex = c (b) y = log cex (c) y = log(ex+c) (d) ex+y = c8) dt

dp = ket (k xU kh)

(a) c - tek

= p (b) p = ket + c(c) t = log k

pc (d) t = logc p

9) (x2 - y2) dy = 2xy dx vw tifbfG rkgho y = vxvd uLbghGJ rkghL flitf vJthf

khW?

(a)

3

21vv

v

++

dv = x

dx (b) )1(1

2

2

vv

v

+

dv = x

dx

(c) 12 vdv

= x

dx (d) 21 vdv+

= x

dx

38

10) x dxdy

= y + 22 yx + vw rkgho y = vx vd ul/rkghL flitf vJthf khW?

(a) 12 vdv

= x

dx (b) 12 +vvdv

= x

dx

(c) 12 +vdv

= x

dx (d) 21 vvdv

= x

dx

11) dxdy

+ Py = 0 vw toila rkgho (P MdJ x, rh)(a) y dxe P = c (b) y dxP = c(c) x dxe P = y (d) y = cx

12) dydx

+ Px = Q vw toila rkgho (P kW Qvgd y , rh)(a) y = Q dxe P dy +c (b) y dxe P = Q dxe P dx+c(c) x dye P = Q dye P dy +c (d) x dye P = Q dxe P dx +c

13) x dxdy

y = ex - bjifL fhu

(a) logx (b) xe1 (c)

x1 (d)

x1

14) (1 + x2)

dxdy

+ xy = (1 + x2)3 - bjifL fhu

(a) 21 x+ (b) log (1 + x2) (c) etan-1x (d) log(tan-1x)

15)x

ydxdy 2

+ = x3 -vw rkgho bjifL fhu

(a) 2 log x (b) 2xe (c) 3 log(x2) (d) x2

16) (D2 D) y = ex -vw tifbfG rkgho u rh(a) A + B ex (b) (Ax + B)ex (c) A + Bex (d) (A+Bx)e-x

17) (D2 2D + 1)y = e2x -vw tifbfG rkgho urh

(a) Aex + Bex (b) A + Bex (c) (Ax + B)ex (d) A+Bex

39

18) 22

dxyd

5 dxdy

+ 6y = e5x -vw tifbfG rkghow bjhif

(a) 65 x

e (b) ! 25 x

xe (c) 6e5x (d) 255 x

e

19) 22

dxyd

6 dxdy

+ 9y = e3x -vw tifbfG rkgho wbjhif

(a) ! 23 x

e (b) ! 232 x

ex (c) ! 23 x

xe (d) 9e3x

20) 22

dxyd

y = 0 vw rkgho

(a) (A + B)ex (b) (Ax + B)ex (c) Aex +

Bxe

(d) (A+Bx)ex

40

7.1 ,ilbrUf,ilbrUf,ilbrUf,ilbrUf,ilbrUf (INTERPOLATION),ilbrUf vgJ mltiz bfhLfgLs

tufis bfhL/ bfhLfglhj xU kidfhQw fiyahF. mjhtJ bfhLfgl rhkfis bfhL mkfSF ,ilgl kfisfhgijnah myJ utijnah ,ilbrUf vnwh.tU mltiz xU efu gJ MLF xU KiwfzlgL kf bjhif tu bfhLfgLsJ.

ML x : 1910 1920 1930 1940 1950kfbjhif f(x) : 12 15 20 27 39 (Muf)

nkny bfhLfgLs tufis bfhL 1914,1923, 1939, 1947 Ma MLf kf bjhifiafhQ Kiw ,ilbrUf vdgL. 1955, 1960 MaMLf kf bjhifia fhQ Kiw w brUfvdgL.

,il brUfiy fhz tUtdtiw fUbfhf :

(i) f(x)- kf VW tirnyh myJ ,wFtirnyh ,Uf ntL.

(ii) f(x)- kf uhf ,Uf ntL. mjhtJx- VjhtJ ,uL kfSilna f(x)-kf O Vwnkh myJ O ,wfnkh,Uf TlhJ.

tU Kiwf ,ilbrUfiy fhzyh :

1) tiugl Kiw/ 2) ,afj Kiw

ilbrUf kWnenfhL bghUJj 7

41

7.1.1 tiugl Kiw ,ilbrUf fhztiugl Kiw ,ilbrUf fhztiugl Kiw ,ilbrUf fhztiugl Kiw ,ilbrUf fhztiugl Kiw ,ilbrUf fhz(Graphic method of interpolation)y = f(x) vf. x- kfSF mjF Vw y-

kfSF Vg tiugl fis Ff. ,jtiu gl _y bfhLfgl x-F Vw y- kigfhzyh.

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 1tU tufis bfhLtU tufis bfhLtU tufis bfhLtU tufis bfhLtU tufis bfhL 1986 MMMMM

Mo kf bjhifia tiugl _yMo kf bjhifia tiugl _yMo kf bjhifia tiugl _yMo kf bjhifia tiugl _yMo kf bjhifia tiugl _ykLfkLfkLfkLfkLf.

MLMLMLMLML : 1960 1970 1980 1990 2000kfbjhifkfbjhifkfbjhifkfbjhifkfbjhif: 12 15 20 26 33(MufMufMufMufMuf) :

tiuglUJ 1986 Mo kf bjhif 24 MuMF.

1960 1970 1980 1990 2000 2010ML

34323028262422201816141210

k

f

bj

hif

(M

u

f

)

1986(1960, 12)

(1970, 15)

(1980, 20)

(1986, 24)

(1990, 26)

(2000, 33)

x

y

42

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 2tiugl _y/tiugl _y/tiugl _y/tiugl _y/tiugl _y/ x = 27 Mf ,UFbghGJMf ,UFbghGJMf ,UFbghGJMf ,UFbghGJMf ,UFbghGJ

y- kig fhf kig fhf kig fhf kig fhf kig fhf.x : 10 15 20 25 30y : 35 32 29 26 23

:

x = 27 Mf ,UFbghGJ y = 24.8 MF.

7.1.2 ,ilbrUfYfhd ,afj Kiwf,ilbrUfYfhd ,afj Kiwf,ilbrUfYfhd ,afj Kiwf,ilbrUfYfhd ,afj Kiwf,ilbrUfYfhd ,afj Kiwf,ilbrUf fhgjfhd fj Kiwf gy csd.

mtW tU y Kiwfis fhngh.

(i) lkhd ntWghLf (Finite differences)(ii) nfh-l Nuf (Gregory-

Newtons formulae)(iii) ,yuh Nu (Lagranges formula)

10 15 20 25 30

35343332313029282726252423

24.8

27

(30, 23)

(27, 24.8)(25, 26)

(20, 29)

(15, 32)

(10, 35)

x

y

43

7.1.3 lkhd ntWghLflkhd ntWghLflkhd ntWghLflkhd ntWghLflkhd ntWghLfx0, x1, x2, ... xn vw rh khfis (arguments)

y0, y1, y2, . . . , yn vw rhgyfis (entries)vLJbfhnth. ,F y = f(x) vgJ ,ilbrUfgagLjgL xU rh MF.

x- kf VW tir ,Ugjhf kW mitrk ,ilbtf ,Ugjhf vLJbfhnth. rk,ilbtf s h vngh.

x0, x0 + h, x0 + 2h, ... x0 + nh vgd x- kfvngh. mtFa rhgyf f(x0), f(x0+h), f(x0 + 2h),..., f(x0 + nh) vgdthFK nehF ntWghL bra K nehF ntWghL bra K nehF ntWghL bra K nehF ntWghL bra K nehF ntWghL bra (Forward differenceoperator) x- vj xU kF/ KnehF bra (bllh)it

f(x) = f(x+h) - f(x) vd tiuaWfyh. Fghf, y0 = f(x0) = f(x0+h) f(x0) = y1y0

f(x), [f(x+h)], [f(x+2h)], ... vgd f(x)- KjiyntWghLf MF.

2 f(x) = [{f(x)}]= [f(x+h) f(x)]= [f(x+h)] [f(x)]= [f(x+2h) f(x+h)] [f(x+h) f(x)]= f(x+2h) 2f (x+h) + f(x).

2 f(x), 2 [f(x+h)], 2 [f(x+2h)] ... vgd f(x)-,ulh iy ntWghLf MF.

,nj ngh 3 f (x ) , 4 f(x), . . .n f(x), . . . vgdtiuaWfgLwd.

44

nehF ntWghL bra nehF ntWghL bra nehF ntWghL bra nehF ntWghL bra nehF ntWghL bra (Backward differenceoperator)

x- vj xU kF/ nehF ntWghL bra(beyh) it

f(x) = f(x) f(x h) vd tiuaWfyh. Fghf, y

n= f(x

n) = f(x

n) f(x

n h) = y

ny

n1

f(x), [f(x+h)], [f(x+2h)], ... vgd f(x)- KjiyntWghLf MF.

2 f(x) = [{f(x)}] = [f(x) f(xh)]= [f(x)] [f(x h)]= f(x) 2f(x h) + f(x2h)

2 f(x), 2 [f(x+h)], 2 [f(x+2h)] ... vgd f(x) ,ulhiy ntWghLf MF.

,nj ngh 3 f (x ) , 4 f(x), . . .n f(x), . . . vgdtiuaWfgLwd.

,lbga bra ,lbga bra ,lbga bra ,lbga bra ,lbga bra (Shifting operator)x- vj xU kF/ ,lbga bra E I

E[f(x)] = f(x+h) vd tiuaWfyh.Fghf, E(y0) = E[f(x0)] = f(x0+h) = y1nkY, E2 [f(x)] = E[E{f(x)] = E[f(x+h)] = f(x+2h),njngh E3[f(x)] = f(x+3h)bghJthf En [f(x)] = f(x+nh) FFFFF E F ,ilna cs bjhlF ,ilna cs bjhlF ,ilna cs bjhlF ,ilna cs bjhlF ,ilna cs bjhltiuaiwgo f(x)= f(x+h) f(x)

= E f(x) f(x)f(x) = (E 1) f(x)

= E 1(m-J) E = 1+

45

KofKofKofKofKof

1) kh rh ntWghLf akhf ,UF.2) f(x) vgJ n M go gYW nfhit v f(x) n

M iy ntWghLf khahF kW n+1 f(x) = 0.vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 3

tU tufis bfhL LgltU tufis bfhL LgltU tufis bfhL LgltU tufis bfhL LgltU tufis bfhL LglcWig fhf.cWig fhf.cWig fhf.cWig fhf.cWig fhf.

x : 1 2 3 4f(x) : 100 -- 126 157

:

f(x)- _W kf bfhLfgoUgjh/ f(x) xU,ulhgo gYW nfhit vd vLJ bfhnth.

vdnt _wh iy ntWghLf akhF.

3 [f(x0)] = 0myJ 3(y0) = 0

(E 1)3 y0= 0 ( = E 1)(E3 3E2 + 3E 1) y0 = 0

y3 3y2 + 3y1 y0 = 0

157 3(126) + 3y1 100 = 0 y1 = 107

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 4bfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhL 1962

kWkWkWkWkW 1965 M MLfSfhd cgfisM MLfSfhd cgfisM MLfSfhd cgfisM MLfSfhd cgfisM MLfSfhd cgfisfhf.fhf.fhf.fhf.fhf.

MLMLMLMLML : 1961 1962 1963 1964 1965 1966 1967cgcgcgcgcg : 200 -- 260 306 -- 390 430(lf)(lf)(lf)(lf)(lf)

46

:

f(x) - IJ kf bfhLfgoUgjh f(x) xUehfhgo gYW nfhit vd vLJbfhnth. vdntIjh iy ntWghLf akhF.

5 [f(x0)] = 0(m-J) 5 (y0) = 0 (E 1)5 (y0) = 0(m-J) (E5 5E4 + 10E3 10E2 + 5E 1) y0= 0

y5 5y4 + 10y3 10y2 + 5y1 y0 = 0390 5y4 + 10(306) 10(260) + 5y1 200 = 0

y1 y4 = 130 --------------(1)Ijh iy ntWghLf akhf ,Ugjh/ nkY

5 [f(x1)] = 0(m-J) 5 (y1) = 0(m-J) (E 1)5 y1 = 0

(E5 5E4 + 10E3 10E2 + 5E 1)y1 = 0y6 5y5 + 10y4 10y3 + 5y2 y1 = 0430 5(390) + 10y4 10(306) + 5(260) y1 = 0

10y4 y1 = 3280 ------------(2)rkghLf (1) kW (2) fis FbghGJ ekFilgJ

y1 = 220 kW y4 = 350 1962 kW 1965 M MLf cgf Kiwna220 lf kW 350 lf MF.

7.1.4 nfh-l KnehF Nuijnfh-l KnehF Nuijnfh-l KnehF Nuijnfh-l KnehF Nuijnfh-l KnehF NuijjUF KiwjUF KiwjUF KiwjUF KiwjUF Kiwy = f(x) vgJ n M go gYWnfhit vf. x MdJ

x0, x1, x2, ... xn vw rk ,ilbtY/ VW tirY

47

cs kfis bgWbghGJ y MdJ Kiwna f(x0), f(x1),f(x2)... f(xn) Ma (n+1) kfis milwJ.x1 x0 = x2 x1 = x3 x2 = ... = xn xn-1 = h vf(h xU if v),F f(x0) = y0, f(x1) = y1, ... f(xn) = yn

,bghGJ f(x) I tUkhW vGjyh.f(x) = a0 + a1 (x x0) + a2(xx0)(xx1) + ...

+an(xx0) (xx1)... (xxn-1) ----------------(1)

x = x0 v/ (1) f(x0) = a0 myJ a0 = y0

x = x1 v, (1) f(x1) = a0 + a1 (x1 x0) (m-J) y1 = y0 + a1 h

a1 = hyy 01

a1 =

hy0

x = x2 v, (1) f(x2) = a0 + a1(x2 x0) + a2(x2 x0) (x2 x1)y2 = y0 +

hy0

(2h) + a2 (2h) (h) 2h2 a2 = y2 y0 2y0

= y2 y0 2(y1 y0)= y2 2y1 + y0 = 2 y0

a2 =

20

2

! 2

hy

,nj ngh

a3 = 30

3

! 3

hy

, a4 = 40

4

! 4

hy

,..., an = n

n

hny

! 0

vd ilF.

a0, a1, ..., an vgj kfis (1) ulf(x) = y0 + h

y0 (x - x0) +

20

2

! 2

hy

(x x0) (x x1) + ...

+ n

n

hny

! 0 (x x0) (x x1) ... (x xn-1) ---------(2)

48

u = hxx 0 vd vLJbfhlh

x x0 = hu x x1 = (x x0) (x1 x0) = hu h = h(u1) x x2 = (x x0) (x2 x0) = hu 2h = h(u2) x x3 = h (u - 3)

bghJthf/

x xn-1 = h{u (n1)}

Mjyh (2) I tUkhW vGjyh.

f(x) = y0 +

! 1u

y0 + ! 2)1( uu 2y0 + ...

+!

)1)...(2)(1(n

nuuuu ny0u = h

xx 0. ,Jnt nfh-l KnehF

NukhF.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 5x : 0 1 2 3 4y : 176 185 194 202 212

vd bfhLfgoUFbghGJ/ vd bfhLfgoUFbghGJ/ vd bfhLfgoUFbghGJ/ vd bfhLfgoUFbghGJ/ vd bfhLfgoUFbghGJ/ x = 0.2 v v v v v ykig fhf.kig fhf.kig fhf.kig fhf.kig fhf.

:

0.2 vgJ (x0, x1) vw Kj ,ilbt csJ.(m-J) (0, 1)F ,il csJ. vdnt ,ilbrUfYfhdnfh-l KnehF Nuij gagLJnth.IJ kf bfhLfgoUgjh ,ilbrUfYfhdNu/

y = y0 +

! 1u

y0 + ! 2)1( uu 2 y0 + ! 3

)2)(1( uuu 3y0

49

+ ! 4)3)(2)(1( uuuu 4y0 ,F u = h

xx 0

h = 1, x0 = 0 kW x = 0.2 MF.

u =

102.0

= 0.2

KnehF ntWghL mltiz:

x y y 2y 3y 4y0 176 91 185 9 0 -12 194 8 -1 3 43 202 10 24 212

y = 176 +

! 12.0

(9) + ! 2)12.0(2.0 (0)

+

! 3)22.0)(12.0)(2.0(

(-1) +

! 4)32.0)(22.0)(12.0)(2.0( (4)

= 176 + 1.8 0.048 0.1344= 177.6176

x = 0.2 v, y = 177.6176 MF.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 6 y75 = 2459, y80 = 2018, y85 = 1180 kWkWkWkWkW y90 = 402 vvvvv y82 I fhf.I fhf.I fhf.I fhf.I fhf.

:

bfhLfgl tufis tUkhW vGjyh:

x : 75 80 85 90y : 2459 2018 1180 40282 vgJ (80, 85) vw ,ilbt csJ. vdnt

,ilbrUfYfhd nfh-l KnehF

50

Nuij gagLJnth. ehF kf bfhLfgoUgjh ,ilbrUfYfhd Nu/

y = y0 +

! 1u

y0 + ! 2

)1( uu 2 y0 + ! 3

)2)(1( uuu 3y0

,F u = hxx 0

h = 5, x0 = 75 x = 82

u =

57582

=

57

= 1.4

KnehF ntWghL mltiz :

x y y 2y 3y75 2459

-44180 2018

-838 -397 45785 1180

-778 6090 402

y = 2459 + ! 14.1

(-441) + ! 2)14.1(4.1 (-397)

+ ! 3)24.1)(14.1(4.1 (457)

= 2459 617.4 111.6 25.592 x = 82 v/ y = 1704.408

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 7bfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhL

e1.75 - - - - - kig fhf. kig fhf. kig fhf. kig fhf. kig fhf.

x : 1.7 1.8 1.9 2.0 2.1ex : 5.474 6.050 6.686 7.389 8.166

:

IJ kf bfhLfgoUgjh/ ,ilbrUfYfhd Nu/

51

yx = y0 +

! 1u

y0 + ! 2)1( uu 2 y0 + ! 3

)2)(1( uuu 3y0

+ ! 4)3)(2)(1( uuuu 4y0

,F u = hxx 0

h = 0.1, x0 = 1.7 x = 1.75

u =

1.07.175.1

=

1.005.0

=0.5

KnehF ntWghL mltiz :

x y y 2y 3y 4y1.7 5.474

0.5761.8 6.050 0.060 0.0071.9 6.686

0.6360.067 0.007 0

2.0 7.3890.703

0.074

2.1 8.1660.777

y = 5.474 + ! 15.0 (0.576) + ! 2

)15.0(5.0 (0.06)

+

! 3)25.0)(15.0(5.0

(0.007)= 5.474 + 0.288 0.0075 + 0.0004375

x = 1.75 v/ y = 5.7549375

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 8bfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhL

80br. UJbr. UJbr. UJbr. UJbr. UJ 90br. tiu cauKs khztfbr. tiu cauKs khztfbr. tiu cauKs khztfbr. tiu cauKs khztfbr. tiu cauKs khztfvifid fhf.vifid fhf.vifid fhf.vifid fhf.vifid fhf.

cau(br..)cau(br..)cau(br..)cau(br..)cau(br..) x : 40-60 60-80 80-100 100-120 120-140khztfkhztfkhztfkhztfkhztf y : 250 120 100 70 50vifvifvifvifvif

52

:KnehF ntWghL mltiz

x y y 2y 3y 4y ( F cnsh)

60 25080 370 120 -20

-10100 470 100 -30 10 20120 540 70 -20140 590 50

90brF Fiwthd cau cila khztfvifia fhngh.

,F x = 90 u =

hxx 0

=

206090

= 1.5

y(90) = 250 +(1.5)(120) +

! 2)15.1)(5.1(

(20)

+

! 3)25.1)(15.1)(5.1(

(-10)+

! 4)35.1)(25.1)(15.1)(5.1( (20)

= 250 + 180 7.5 + 0.625 + 0.46875= 423.59 ~ 424

80br..UJ 90br.. tiu cauKs khztfvif y(90) y(80)

424 370 = 54.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 9bfhLfgLs tufis bfhL %bfhLfgLs tufis bfhL %bfhLfgLs tufis bfhL %bfhLfgLs tufis bfhL %bfhLfgLs tufis bfhL %.30

UJ %UJ %UJ %UJ %UJ % .35 tiu T bgWgtftiu T bgWgtftiu T bgWgtftiu T bgWgtftiu T bgWgtfvifia fhf.vifia fhf.vifia fhf.vifia fhf.vifia fhf.

TTTTT x : 20-30 30-40 40-50 50-60egfegfegfegfegf y : 9 30 35 42vifvifvifvifvif

53

:

KnehF ntWghL mltiz

x y y 2y 3y( F Fiwthf)

30 940 39 30 550 74 35 7 2

60 116 42

%.35F Fiwthf T bgWgtf vifiafhngh.

,F x = 35 , u =

hxx 0

=

103035

= 0.5

nfh-l KnehF Nugo/

y(35) = 9+

1)5.0(

(30) + ! 2)15.0)(5.0( (5)

+

! 2)25.0)(15.0)(5.0( (2)

= 9 + 15 0.6 + 0.1= 24 (njhuhakhf)

%.30 UJ %.35 tiu T bgWgtf vify(35) y(30) 24 9 = 15.7.1.5 nfh-l nehF Nuijnfh-l nehF Nuijnfh-l nehF Nuijnfh-l nehF Nuijnfh-l nehF Nuij

jUF KiwjUF KiwjUF KiwjUF KiwjUF Kiw

y = f(x) vgJ n M go gYWnfhit vf. x MdJx0, x1, x2, ... xn vw rk ,ilbtY/ VW tirYcs kfis bgWbghGJ y MdJ Kiwna f(x0), f(x1),f(x2)... f(xn) Ma (n+1) kfis milwJ.x1 x0 = x2 x1 = x3 x2 = ... xn xn-1 = h vf (h xU if v),F f(x) vgij tUkhW vGjyh.

54

f(x) = a0 + a1(xxn) + a2(xxn) (xxn-1) + ...+ a

n(xx

n) (xx

n-1) ... (xx1) -----------(1) x = x

n v , (1) f(x

n) = a0 (m-J) a0 = yn

x = xn1 v, (1) f(xn1)= a0 + a1(xn1xn)

myJ yn1 = yn + a1 (h) myJ a1=

hyy nn 1

a1 =

hyn

x = xn2 v, (1)

f(xn2) = a0 + a1 (xn2 xn) + a2 (xn2 xn) (xn2 xn1)

yn2 = yn +

hyn

(2h) + a2 (2h) (h)2h2a2 = (yn2 yn) + 2yn

= yn2 yn + 2(ynyn1)

= yn2 2yn1 + yn = 2yn

a2 =

2

2

! 2 hyn

,njnghW tUtdtiw eh bgwyh.

a3 = 3

3

! 3 hyn

, a4 = 4

4

! 4 hyn

... an =

! nyn

n

f(x) = yn + h

yn (xxn) +

2

2

! 2 hyn

(xxn)(xx

n1) + ...

+ ! nyn

n (xxn) (xx

n1) ... (xx1) ------------(2)

nkY/ u = hxx n

v/

xxn

= hu

xxn1 = (xxn) (xnxn1) = hu + h = h(u+1)

xxn2 = (xxn) (xnxn2) = hu + 2h = h(u+2)

xxn3 = h(u+3)

55

bghJthf/

xxnk = h(u+k)

Mjyh (2) I tUkhW vGjyh.

f(x) = yn +

! 1u

yn+ ! 2

)1( +uu 2y

n + ...

+ ! )}1()...{1(

n

nuuu ++ ny

n ,F u =

hxx n

,Jnt nfh-l nehF NukhF.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 10xU efu kf bjhif tu nHxU efu kf bjhif tu nHxU efu kf bjhif tu nHxU efu kf bjhif tu nHxU efu kf bjhif tu nH

bfhLfgLsJ.bfhLfgLsJ.bfhLfgLsJ.bfhLfgLsJ.bfhLfgLsJ.

MLMLMLMLML x : 1961 1971 1981 1991 2001kfbjhifkfbjhifkfbjhifkfbjhifkfbjhif y : 46 66 81 93 101 (MufMufMufMufMuf)

nfh-l Nuij gagLnfh-l Nuij gagLnfh-l Nuij gagLnfh-l Nuij gagLnfh-l Nuij gagL 1995M Mo kf bjhifia fhf.M Mo kf bjhifia fhf.M Mo kf bjhifia fhf.M Mo kf bjhifia fhf.M Mo kf bjhifia fhf.

:

1995 vgJ fil ,ilbt (1991, 2001) csJ.vdnt nfh-l nehF NuijgagLJnth. IJ kf bfhLfgoUgjh,ilbrUfYfhd Nu/

y = y4 +

! 1u

y4 + ! 2)1( +uu 2 y4 + ! 3

)2)(1( ++ uuu 3y4

+ ! 4)3)(2)(1( +++ uuuu 4y4 ,F u = h

xx 4

h = 10, x4 = 2001 x = 1995

u =

1020011995

= 0.6

56

nehF ntWghL mltiz :

x y y 2y 3y 4y1961 46

201971 6615

-5 21981 81

12-3

-1 -31991 93

8-4

2001 101101101101101

y = 101 +

! 1)6.0(

(8) +

! 2)16.0)(6.0( +

(4)

+

! 3)26.0)(16.0)(6.0( ++

(1)

+

! 4)36.0)(26.0)(16.0)(6.0( +++

(3)= 1014.8+0.48+0.056+0.1008

y = 96.8368 1995 M Mo kf bjhif 96.837 Muf.vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 11

tU tufis bfhLtU tufis bfhLtU tufis bfhLtU tufis bfhLtU tufis bfhL 58 tatatatataKailaToa fhL KailaToa fhL KailaToa fhL KailaToa fhL KailaToa fhL (policy) x x x x xfhL bjhifia fhL bjhifia fhL bjhifia fhL bjhifia fhL bjhifia (premium) fhf. fhf. fhf. fhf. fhf. tttttaJaJaJaJaJ x : 40 45 50 55 60fhLfhLfhLfhLfhL y : 114.84 96.16 83.32 74.48 68.48bjhifbjhifbjhifbjhifbjhif

:IJ kf bfhLfgoUgjh ,il

brUfYfhd Nu/

y = y4 +

! 1u

y4 +...+ ! 4)3)(2)(1( +++ uuuu 4y4

,F u = 56058

= 0.4

57

nehF ntWghL mltiz :

x y y 2y 3y 4y40 114.84

-18.6845 96.16

-12.845.84

-1.8450 83.32

-8.844.00

-1.160.68

55 74.48-6.00 2.84

60 68.4868.4868.4868.4868.48

y = 68.48 +

! 1)4.0(

(-6) +

2)6.0)(4.0(

(2.84)

+

6)6.1)(6.0)(4.0(

(-1.16) +

24)6.2)(6.1)(6.0)(4.0(

(0.68)= 68.48 + 2.4 0.3408 + 0.07424 0.028288

y = 70.5851052 (m-J) y ~ 70.59 58 ta Kaila Toa fhL x fhLbjhif 70.59vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 12

bfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLbfhLfgLs tufis bfhLx = 4.5 F F F F F y kig fhf. kig fhf. kig fhf. kig fhf. kig fhf.

x : 1 2 3 4 5 y : 1 8 27 64 125

:

IJ kf bfhLfgoUgjh ,ilbrUfYfhd Nu/

y = y4 +

! 1u

y4 +...+ ! 4)3)(2)(1( +++ uuuu

4y4

,F u = hxx 4

u =

155.4

= 0.5

58

nehF ntWghL mltiz :

x y y 2y 3y 4y1 1 72 8 19 12 63 27 37 18 6 04 64 61 245 125125125125125

y = 125+

1)5.0(

(61)+

2)5.0)(5.0(

(24) +

6)5.1)(5.0)(5.0(

(6) x = 4.5 v/ y = 91.125

7.1.6 ,yuh Nu,yuh Nu,yuh Nu,yuh Nu,yuh Nuy = f(x) vgJ n M go gYWnfhit vf. x MdJ

x0, x1, x2, ... xn vw VW tir cs kfisbgWbghGJ y MdJ Kiwna f(x0), f(x1), f(x2)... f(xn) Ma(n+1) kfis milwJ (x vgJ rk ,ilbt,Uf ntoa mtaiy).

,F f(x0) = y0, f(x1) = y1, ..., f(xn) = yn.,yuh Nu

f(x) = y0

))...()(())...()((

02010

21

n

n

xxxxxx

xxxxxx

+ y1

))...()(())...()((

12101

20

n

n

xxxxxx

xxxxxx

+ ... + yn

))...()(())...()((110

110

nnnn

n

xxxxxx

xxxxxx

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 13tU mltiz tU mltiz tU mltiz tU mltiz tU mltiz x = 42 MfMfMfMfMf

,UFbghGJ ,UFbghGJ ,UFbghGJ ,UFbghGJ ,UFbghGJ y- kig ,yuh kig ,yuh kig ,yuh kig ,yuh kig ,yuhNuij gagL fhf.Nuij gagL fhf.Nuij gagL fhf.Nuij gagL fhf.Nuij gagL fhf.

59

x : 40 50 60 70y : 31 73 124 159

:

bfhLfgl tufUJ/

x0 = 40, x1 = 50, x2 = 60, x3 = 70 kW x = 42y0 = 31, y1 = 73, y2 = 124, y3 = 159

,yuh Nuij gagLj ekF ilgJ/

y = y0 ))()(())()((302010

321

xxxxxx

xxxxxx

+ y1

))()(())()((312101

320

xxxxxx

xxxxxx

+ y2

))()(())()((321202

310

xxxxxx

xxxxxx

+ y3

))()(())()((231303

210

xxxxxx

xxxxxx

y(42) = 31 )30)(20)(10()28)(18)(8(

+ 73 )20)(10)(10()28)(18)(2(

+124

)10)(10)(20()28)(8)(2(

+159

)10)(20)(30()18)(8)(2(

= 20.832 + 36.792 27.776 + 7.632 y = 37.48

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 14tU mltizia bfhLtU mltizia bfhLtU mltizia bfhLtU mltizia bfhLtU mltizia bfhL

,yuh Nuij gagL,yuh Nuij gagL,yuh Nuij gagL,yuh Nuij gagL,yuh Nuij gagL x = 4 MfMfMfMfMf,UFbghGJ ,UFbghGJ ,UFbghGJ ,UFbghGJ ,UFbghGJ y- kig fhf. kig fhf. kig fhf. kig fhf. kig fhf.

x : 0 3 5 6 8y : 276 460 414 343 110

60

:

x0 = 0, x1 = 3, x2 = 5, x3 = 6, x4 = 8 kW x = 4y0 = 276, y1 = 460, y2 = 414, y3 = 343, y4 = 110

vd bfhLfgLsJ,yuh Nuij gagLj ekF ilgJ/

y = y0

))()()(())()()((

40302010

4321

xxxxxxxx

xxxxxxxx

+ y1

))()()(())()()((41312101

4320

xxxxxxxx

xxxxxxxx

+ y2

))()()(())()()((

42321202

4310

xxxxxxxx

xxxxxxxx

+ y3

))()()(())()()((

43231303

4210

xxxxxxxx

xxxxxxxx

+ y4

))()()(())()()((

34241404

3210

xxxxxxxx

xxxxxxxx

= 276)8)(6)(5)(3(

)4)(2)(1)(1(

+ 460)5)(3)(2)(3()4)(2)(1)(4(

+ 414)3)(1)(2)(5()4)(2)(1)(4(

+343)2)(1)(3)(6()4)(1)(1)(4(

+ 110

)2)(3)(5)(8()2)(1)(1)(4(

= 3.066 + 163.555 + 441.6 152.44 + 3.666 y = 453.311

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 15tU mltizia bfhLtU mltizia bfhLtU mltizia bfhLtU mltizia bfhLtU mltizia bfhL

,yuh Nuij gagL,yuh Nuij gagL,yuh Nuij gagL,yuh Nuij gagL,yuh Nuij gagL y(11) kig fhf.kig fhf.kig fhf.kig fhf.kig fhf.

x : 6 7 10 12y : 13 14 15 17

61

:

x0 = 6, x1 = 7, x2 = 10, x3 = 12 kW x = 11y0 = 13, y1 = 14, y2 = 15, y3 = 17 vd bfhLfgLsJ

,yuh Nuij gagLj ekF ilgJ/

= 13

)6)(4)(1()1)(1)(4(

+ 14

)5)(3)(1()1)(1)(5(

+ 15

)2)(3)(4()1)(4)(5(

+17

)2)(5)(6()1)(4)(5(

= 2.1666 4.6666 + 12.5 + 5.6666 y = 15.6666ggggg 7.1

1) nH bfhLfgLs tufis bfhL x = 42 Mf,UFnghJ y- kig tiugl _y fhf.

x : 20 30 40 50y : 51 43 34 24

2) xU efu kf bjhif nH bfhLfgLsJML x : 1940 1950 1960 1970 1980 1990kfbjhif y : 20 24 29 36 46 50(,yrf)1976 Mo kf bjhifia tiugl _y fhf.

3) bfhLfgLs tufis bfhL f(3) I fhf.x : 1 2 3 4 5f(x) : 2 5 - 14 32

4) bfhLfgLs tufis bfhL Lgl vizfhf.x : 0 5 10 15 20 25y : 7 11 14 -- 24 32

5) bfhLfgLs tufis bfhL 2000 M MoVWkia kLf.

ML x : 1999 2000 2001 2002 2003VWk y : 443 -- 369 397 467(lf)

62

6) bfhLfgLs tufis bfhL x = 145 Mf,UFbghGJ y - kig nfh-lNuij gagL fhf.

x : 140 150 160 170 180y : 46 66 81 93 101

7) bfhLfgLs tufis bfhL y(8) - kignfh-l Nuij gagL fhf.

x : 0 5 10 15 20 25y : 7 11 14 18 24 32

8) 1975 M Mo kf bjhifia nfh-lNuij gagL fzLf.

tUl : 1961 1971 1981 1991 2001kfbjhif : 98572 132285 168076 198690 246050

9) bfhLfgLs tufis bfhL l 96 myFfcs tl guig nfh-l NuijgagL fhf.

l x : 80 85 90 95 100gu y : 5026 5674 6362 7088 7854

10) nfh-l Nuij gagL x = 85 Mf,UFbghGJ y - kig fhf.x : 50 60 70 80 90 100y : 184 204 226 250 276 304

11) nfh-l Nuij gagL y(22.4) -kig fhf.

x : 19 20 21 22 23y : 91 100 110 120 131

12) bfhLfgLs tufis bfhL ,yuhNuij gagL y(25) - kig fhf.x : 20 30 40 50y : 512 439 346 243

63

13) f(0) = 5, f(1) = 6, f(3) = 50, f(4) = 105 v ,yuhNuij gagL f(2) - kig fhf.

14) bfhLfgLs tufis bfhL/ ,yuhNuij gagL x = 5 v/ y - kig fhf.x : 1 2 3 4 7y : 2 4 8 16 128

7.2 nenfhL bghUJjnenfhL bghUJjnenfhL bghUJjnenfhL bghUJjnenfhL bghUJjbghJthf gy Jiwf ,uL (myJ mjF nkgl)

khfSF ,ilna cs bjhlf FJ Muha ntoamta csJ.

cjhuzkhf xU FHij vilahdJ mj taJlbjhlilaJ; xU bghU iyahdJ mbghUnjitnahL bjhlilaJ; xU thfd guhkbrythdJ mJ gagLjgl fhynjhL bjhlilaJ.

7.2.1 jw tiugljw tiugljw tiugljw tiugljw tiugl (Scatter diagram),U khf x kW y vgd xU M taJ kW

vilia FwJ vd vLJ bfhlh/ x1, x2, x3, ...x

n vgd n Mf taij y1, y2, y3, ... yn vgd

Kiwna mtf viliaFwd vngh. (x1, y1),(x2, y2), (x3, y3), ... (xn, yn)vw fis xU brtfyM a b j h i y f FLnth. ,thWFLtjh tiuglilF f fzijjw tiugl vngh.

jw tiugl _y bfhLfgl kfSfhdfis mQ tUkhW xU uhd tistiu ,UfTLvgij eh fhzyh. ,jifa tistiuia mQtUw tistiu vngh. nkny cs gl 7.1

y

xgl 7.1

64

bfhLfgl kf xU nenfhil mQ tUwdvgij kW ,uL khfSilna xU neabjhl ,Ugij czuyh.

7.2.2 W tf bfhif W tf bfhif W tf bfhif W tf bfhif W tf bfhif (Principle of least squares)bghJthf bfhLfgl kfSF xWF

nkgl tistiuf bghUJtJ ngh njhW. vdntnenfhLf tiubghGJ f wj bghUjkhd xUnenfhoFa tiuaiwia ftd bfhtJmtakhF.

kf (x1, y1), (x2, y2),(x3, y3), ... (xn, yn) vgdtiwfshf vLJ bfhnth.bfhLfgl x = x1 vwkF Vw y- kFtistiu C- _y ilfToa mnj y - kF a h r , U f T L .(gl 7.2)

,j ahrij d1 vf. d1 I yf myJ iHvd Twyh. ,F d1 vgJ if v/ Fiw v myJakhf ,Ufyh. mnj nghW

x2, x3, ... xn -fSfhdyff Kiwna d2, d3, ... dn vd ilF.

bfhLfgl kfSF Vw f wjbghUjkhd tistiu msit d12, d22, ... dn2-fUJbgwyh.

bfhLfgl kfis mQ tUw midJtistiufY d12 + d22 + d32 +...+ dn2 MdJ vtistiuFW kig bfhLsnjh mtistiuna f wjbghUjkhd tistiu vngh. mthW mQ tUwtistiuahdJ nenfhlhf ,U mjid f wjbghUjkhd nenfhL (line of best fit) vngh.

C

x

y (x2, y2)

(x1, y1)

(xn, y

n)

dnd2

d1

gl 7.2

65

7.2.3 W tf bfhif _y ,a iyW tf bfhif _y ,a iyW tf bfhif _y ,a iyW tf bfhif _y ,a iyW tf bfhif _y ,a iyrkghLfis jUjrkghLfis jUjrkghLfis jUjrkghLfis jUjrkghLfis jUj

bfhLfgl (x1, y1), (x2, y2), ... (xn, yn) vw nfSF bghUJ nenfho rkghL

y = ax + b ------------(1)a kW b -f

btntW kfSF (1)MdJ nenfhLFLg xiw FF.(1)-F wjjhfbghUjkhdjhf csa kW b f kfiseh fhzntL. ,kfis fhz Wtf bfhifidgagLJnwh.

Pi (xi, yi) vgJ jw tiugl (gl 7.3) bghJthdxU vf. PiM I x-mRF brFjhf/ y = ax + b IHi btLkhW tiuf. Hi x mR bjhiy xi kW y -mR bjhiy axi + b MF.

PiHi = PiM - HiM= yi (axi +b) vgJ yi - yf MF.

W tf bfhifgo a kW b fkfis fhz ntL. vdnt/

E = n

i 1= PiHi

2 =

n

i 1= [yi (axi + b)]

2 vgJ Wk kig

bgw ntL.

bgWk myJ Wk kfhd gjidfgo akW b ,tiw bghWJ E - gF tif bfGfjjna akhF.

a

E= 0 2

n

i 1= xi[yi (axi + b)] = 0

y

xMO

Pi(xi, yi)

Hi(xi , axi +b)

gl 7.3

66

a n

i 1=

xi2

+ bn

i 1=

xi =n

i 1=

xi yi -------------(2)

bE

= 0 2n

i 1= [yi (axi + b)] = 0

i.e., yi axi nb = 0

an

i 1=

xi + nb = n

i 1=

yi -------------(3)

rkghLf (2) kW (3) Mait ,aiyrkghLf vdgL. ,aiy rkghLfis gj_y a kW b f kfis fhzyh.

FFFFF

y = a + bx vw to cs rkghL f bghUjkhdnenfhlhf miktjfhd ,aiy rkghLf

na + bxi = yiaxi + bxi2 = xi yi

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 16x = 10, y = 19, x2 = 30, xy = 53 kWkWkWkWkW n = 5

vgdtWF xU nenfhil bghUJf.vgdtWF xU nenfhil bghUJf.vgdtWF xU nenfhil bghUJf.vgdtWF xU nenfhil bghUJf.vgdtWF xU nenfhil bghUJf.

:

f bghUjkhd nenfhL y = ax + b vf.y = ax + nbxy = ax2 + bx 10a + 5b = 19 ------------(1)

30a + 10b = 53 ------------(2)(1) kW (2) fis gj thyhf a = 1.5 kW b = 0.8vd bgwyh.

vdnt f bghUjkhd nenfho rkghL

y = 1.5x + 0.8

67

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 17x = 10, y=16.9, x2 = 30, xy = 47.4 kWkWkWkWkW n = 7

vgdtWF jfgo tiuagl fvgdtWF jfgo tiuagl fvgdtWF jfgo tiuagl fvgdtWF jfgo tiuagl fvgdtWF jfgo tiuagl fbghUjkhd nfho bghUjkhd nfho bghUjkhd nfho bghUjkhd nfho bghUjkhd nfho x-m btL Jilm btL Jilm btL Jilm btL Jilm btL Jilfhf.fhf.fhf.fhf.fhf.

:

f bghUjkhd nenfhL y = ax + b vf.,aiy rkghLf

y = ax + nbxy = ax2 + bx 10a + 7b = 16.9 -----------(1)

30a + 10b = 47.4 -----------(2)(1) kW (2) fis gj _y/

a = 1.48 and b = 0.3 vd ilF. f bghUjkhd nenfho rkghL

y = 1.48x + 0.3

vdnt x-m btL JL = 48.10.3

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 18bfhLfgLs tufSF xUbfhLfgLs tufSF xUbfhLfgLs tufSF xUbfhLfgLs tufSF xUbfhLfgLs tufSF xU

nenfhL bghUJf.nenfhL bghUJf.nenfhL bghUJf.nenfhL bghUJf.nenfhL bghUJf.

x : 0 1 2 3 4 y : 1 1 3 4 6

:

f bghUjkhd nenfhL y = ax + b,aiy rkghLf

ax + nb = y ------------(1)ax2 + bx = xy ------------(2)

68

bfhLfgLs tufUJ

x y x2 xy0 1 0 01 1 1 12 3 4 63 4 9 124 6 16 24

10 15 30 43

,kfis rkghLf (1) kW (2) f ul ekFilgJ/

10a + 5b = 15 ------------(3)30a + 10b = 43 ------------(4)

,tiw f/ ekF ilgJ/ a = 1.3 and b = 0.4

f bghUjkhd nenfho rkghL y = 1.3x + 0.4.

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 19bfhLfgLs tufSF xUbfhLfgLs tufSF xUbfhLfgLs tufSF xUbfhLfgLs tufSF xUbfhLfgLs tufSF xU

nenfhL bghUJfnenfhL bghUJfnenfhL bghUJfnenfhL bghUJfnenfhL bghUJfx : 4 8 12 16 20 24y : 7 9 13 17 21 25

:

Mia 21612 +

= 14 vw ,l vLJbfhnth.

ui =

214ix

vf. ,F n = 6

f bghUjkhd nenfhL y = au + b vf.,aiy rkghLf/

au + nb = y ----------(1)au2 + bu = uy ----------(2)

69

x y u u2 uy4 7 -5 25 -358 9 -3 9 -27

12 13 -1 1 -1316 17 1 1 1720 21 3 9 6324 25 5 25 125

bkhjbkhjbkhjbkhjbkhj 92 0 70 130,kfis (1) kW (2) f ul ekF ilgJ/

a = 1.86 kW b = 15.33 f bghUjkhd nenfhL

y = 1.86

214x

+15.33 = 0.93x + 2.31

vLJfhL vLJfhL vLJfhL vLJfhL vLJfhL 20tU tufSF xU nenfhLtU tufSF xU nenfhLtU tufSF xU nenfhLtU tufSF xU nenfhLtU tufSF xU nenfhL

bghUJf.bghUJf.bghUJf.bghUJf.bghUJf.x : 100 200 300 400 500 600y : 90.2 92.3 94.2 96.3 98.2 100.3

:

ui = 50350ix

kW vi = yi 94.2 vf. ,F n = 6.f bghUjkhd nenfhL v = au + b,aiy rkghLf au + nb = v ----------(1)

au2 + bu = uv ----------(2) x y u v u2 uv100 90.2 -5 -4 25 20200 92.3 -3 -1.9 9 5.7300 94.2 -1 0 1 0400 96.3 1 2.1 1 2.1500 98.2 3 4 9 12600 100.3 5 6.1 25 30.5

bkhjbkhjbkhjbkhjbkhj 0 63 70 70.3

70

,kfis rkghLf (1) kW (2) f ulekF ilgJ a = 1.0043 and b = 1.05.f bghUjkhd nenfhL v = 1.0043 u + 1.05

y = 0.02x +88.25

ggggg 7.21) jw tiugl - sFf.2) W tf bfhifia TWf.3) x = 75, y = 115, x2 = 1375, xy = 1875 kW n = 6

v/ xU f wj nenfhL bghUJf.

4) x = 10, y = 25, x2 = 30, xy = 90 kW n = 5 v/xU f bghUjkhd nenfhoid bghU mjrh- kW y-m btL JL Matiw fhf.

5) tU tufis bfhL W tf bfhifiagagL y = ax + b vD nenfhil bghUJf. x : 0 1 3 6 8 y : 1 3 2 5 4

6) 5 khztfis bfhl FG xW xU gF Kmj bgw kbgf tu/

gF K bgw kbgf : 3 4 4 6 8gF bgw kbgf : 4 5 6 8 10W tf bfhifia gagL bghUjkhd nenfhoid fhf.

7) tU tufSF W tf bfhifiagagL f bghUjkhd nenfhoid fhf.x : 100 120 140 160 180 200y : 0.45 0.55 0.60 0.70 0.80 0.85

8) tU tufSF f bghUjkhd nenfhoidfhf. nkY x = 3.5 v y - kig fzLf.x : 0 1 2 3 4y : 1 1.8 3.3 4.5 6.3

71

9) tU tufSF W tf bfhifiagagL f bghUjkhd nenfhL fhf.

gagLjgl MH x : 0 12 24 36 48 (br..f)

ruhr isr y : 35 55 65 80 90 (lf / Vf)

10) khfhf vLJ bfhsgl MW kUJ filfsgu bryf (bkhj bry Gfho) kWfu yhgf (bkhj gid Gfho) tUkhWjugLsJ.

sgu bry : 0.4 1.0 1.3 1.5 2.0 2.8 fu ,yhg : 1.90 2.8 2.9 3.6 4.3 5.4fbghUjkhd nenfhil bghUJf.

11) gJ khztf My ghl goj neu (kf) kWmtf nj bgw kbgf tu tUkhWjugLsJ.

goj neu (kf) x : 4 9 10 12 14 22nj bgw kbgf y : 31 58 65 68 73 91

(i) y = ax + b vw nenfhil bghUJf.(ii) 17 k neu goj xU khzt bgW kbgiz

fhf.

ggggg 7.31) f(x) =

(a) f(x+h) (b) f(x)f(x+h)(c) f(x+h)f(x) (d) f(x)f(xh)

2) E2f(x) =(a) f(x+h) (b) f(x+2h)(c) f(2h) (d) f(2x)

3) E =(a) 1+ (b) 1 (c) + 1 (d) 1

72

4) f(x+3h) =(a) f(x+2h) (b) f(x+3h)-f(x+2h)(c) f(x+3h) (d) f(x+2h) f(x 3h)

5) h = 1 v, (x2) =(a) 2x (b) 2x 1 (c) 2x+1 (d) 1

6) y = ax + b vgJ f wj bghUjkhd nenfhlhfmiktjfhd a kW b vgdtiw fzl njitahd,aiy rkghLf

(a) axi2 + bxi = xiyi kW axi + nb = yi(b) axi + bxi2 = xiyi kW axi2 + nb = yi(c) axi + nb = xiyi kW axi2 + bxi = yi(d) axi2 + nb = xiyi kW axi + bxi = yi

7) f bghUjkhd nenfhlhd y = 5.8 (x-1994) + 41.6 - x= 1997 v/ y k(a) 50 (b) 54 (c) 59 (d) 60

8) X nenfhil bghUJtjfhd IJ kfbfhLfgLsd. nkY x = 0 kW y = 15 MF.,bghGJ f wj bghUjkhd nenfho y -mbtLJL,

(a) 1 (b) 2 (c) 3 (d) 49) y = ax + b vw nenfhil bghUJtjfhd ,aiy

rkghLf 10a +5b = 15 kW 30a + 10b = 43 MF.,bghGJ fbghUjkhd nenfho rh-/

(a) 1.2 (b) 1.3 (c) 13 (d) 1210) n f (x, y) I W tf Kiw y = ax + b vD

nenfho bghUJbghGJ 4 = 4a + b kW xy = 120a+ 24b vw ,aiy rkghLf ilwd v/ n =(a) 30 (b) 5 (c) 6 (d) 4

73

8.1 rkth- kh kW fjf rhrkth- kh kW fjf rhrkth- kh kW fjf rhrkth- kh kW fjf rhrkth- kh kW fjf rh(Random variable and probability function)

rkth- khrkth- khrkth- khrkth- khrkth- khrkth kh vgJ TWbt S - J tiuaWf

gLs bk-kila xU rhghF kW ,kh UJ tiuyhd bk-kfis bgW.

8.1.1 jj rkth- khjj rkth- khjj rkth- khjj rkth- khjj rkth- khX vw kh KoW myJ Kowh Mdh

vljf kf bgWkh mkh xU jjrkth khahF.

cjhuzf

(i) xU ehzaij ,UKiw RLjiy xU nrhjidahffUJnth. ,nrhjidTWf s1 = (H, H),s2 = (H, T), s3 = (T, H) kW s4 = (T, T) MF.rkth- kh X : ,UKiw RL bghGJ ilj

jiyf vif

vdnt X(s1) = 2 ; X(s2) = 1 ; X(s3) = 1 ; X(s4) = 0RX = {0, 1, 2}

s vgJ TWbt cs xU cWghF. ,thwhfX(s) vgJ btghL s -ia bjhl bfhl rkth- khX ia Fw bk- vzhF.

X - vyh kfis bfhl fz RX, X - Rfz vd miHfgLwJ.

(ii) xU nrho gfilfis cULtij nrhjidahfbfhnth. vdnt TWbt

fjf gutffjf gutffjf gutffjf gutffjf gutf 8

74

S = {(1, 1) (1, 2) ... (1, 6) . . .

. . .

. . .

(6, 1) (6, 2) ... (6, 6)}rkth- kh X : ,U gfilf J fhQ vf

TLj vgJ MF. vdnt RX = {2, 3, 4, ..., 12}.(iii) xnu neu 3 ehzafis RLtij nrhjidahf

bfhnth.

rkth- kh X: ,nrhjid ilF jiyfvifia Fgjh/ ,F 0, 1, 2, 3 vw kfisX VwJ.

S = {HHH, HHT, HTT, TTT, TTH, THH, HTH, THT} RX = {0, 1, 2, 3}

(iv) xnu neu 4 ehzafis RLw xU rkth-nrhjid/ ilfbgWw jiyfvifia FLif

RX = {0, 1, 2, 3, 4} vgjhF.xU jf xbthU gf fhzgL

mRiHf vif/ xU Wtd bjhiynggahsuh bgwgL bjhiyng miHf vifMait jj rkth- khfSfhd ycjhuzfshF.

8.1.2 jj rkth- kh fjf rhjj rkth- kh fjf rhjj rkth- kh fjf rhjj rkth- kh fjf rhjj rkth- kh fjf rhkW fjf gutkW fjf gutkW fjf gutkW fjf gutkW fjf gut

X vw jj rkth- kh bgW kf x1, x2, x3...vf. p(xi) = P[X = xi] vwthW cs rh p MdJ

(i) p(xi) > 0 i = 1, 2, ...(ii)

i p(xi) = 1

75

vw gjidfis iw br-kh/ p fjf rhmyJ fjf k rh vW miHfgLwJ.

(x i, p(x i))vw vyh nrhof bjhF X -fjf gut MF.


,U ehzafis RLw nrhjidia,U ehzafis RLw nrhjidia,U ehzafis RLw nrhjidia,U ehzafis RLw nrhjidia,U ehzafis RLw nrhjidiafUJnth. fUJnth. fUJnth. fUJnth. fUJnth. X vw rkth-kh nrhjidvw rkth-kh nrhjidvw rkth-kh nrhjidvw rkth-kh nrhjidvw rkth-kh nrhjidbgwgL jiyf vifia FgjhfbgwgL jiyf vifia FgjhfbgwgL jiyf vifia FgjhfbgwgL jiyf vifia FgjhfbgwgL jiyf vifia Fgjhfbfhnth.bfhnth.bfhnth.bfhnth.bfhnth.

X : 0 1 2

p(xi) : 41

21

41

p(xi) xU fjf k rhgh vd mf.xU fjf k rhgh vd mf.xU fjf k rhgh vd mf.xU fjf k rhgh vd mf.xU fjf k rhgh vd mf. :

(i) ,F xbthU p(xi) > 0 kW(ii) p(xi) = p(0) + p(1) + p(2) vgjid v fhzyh.

= 41

+ 21

+ 41

= 1

vdnt p(xi) fjf krhghF.

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 2,U gfilfis cUL nghJ ilF,U gfilfis cUL nghJ ilF,U gfilfis cUL nghJ ilF,U gfilfis cUL nghJ ilF,U gfilfis cUL nghJ ilF

vf TLj bjhifia vf TLj bjhifia vf TLj bjhifia vf TLj bjhifia vf TLj bjhifia X vw rkth-vw rkth-vw rkth-vw rkth-vw rkth-kh Fgjhf bfhnth. kh Fgjhf bfhnth. kh Fgjhf bfhnth. kh Fgjhf bfhnth. kh Fgjhf bfhnth. X - fjf- fjf- fjf- fjf- fjfgutyhdJgutyhdJgutyhdJgutyhdJgutyhdJ

X : 2 3 4 5 6 7 8 9 10 11 12

p(xi) : 361

362

363

364

365

366

365

364

363

362

361

p(xi) xU fjf k rhgh vd Muh-f?xU fjf k rhgh vd Muh-f?xU fjf k rhgh vd Muh-f?xU fjf k rhgh vd Muh-f?xU fjf k rhgh vd Muh-f?

76

:

(i) ,F xbthU p(xi) > 0 kW

(ii) p(xi) = 361

+ 362

+ 363

+ ........+ 361

= 1 vd ,UgJftfjfJ.

vdnt p(xi) fjf krhghF.8.1.3 F gut rh F gut rh F gut rh F gut rh F gut rh (c.d.f.)

X xU jj rkth- kh vf.

F(x) = P(X < x)=

i p(xi) , i - vyh kfSF xi < x vd

mikkhW TLbjhif fzlgLwJ/ v rhF(x) -ia/ X - F gut rh vd miHnwh.FFFFF : P(a < X < b) = F(b) F(a)vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 3

X vw rkth- kh fjf rhvw rkth- kh fjf rhvw rkth- kh fjf rhvw rkth- kh fjf rhvw rkth- kh fjf rhtUkhW.tUkhW.tUkhW.tUkhW.tUkhW.

X k k k k k, x : 2 1 0 1 2 3 p(x) : 0.1 k 0.2 2k 0.3 k(i) k - kig fhf- kig fhf- kig fhf- kig fhf- kig fhf(ii) X - F gut rhig fzLf.- F gut rhig fzLf.- F gut rhig fzLf.- F gut rhig fzLf.- F gut rhig fzLf.

:

(i)i p(xi) = 1 Mifah

p(2) + p(1) + p(0) + p(1) + p(2) + p(3) = 10.1 + k + 0.2 + 2k + 0.3 + k = 1 k = 0.1vdnt bfhLfgLs fjf rh tUkhW

khWwJ.

77

x : 2 1 0 1 2 3p(x) : 0.1 0.1 0.2 0.2 0.3 0.1

(ii) F gut rh F(x) = P(X < x)x F(x) = P(X < x )2 F(2) = P(X < 2 ) = 0.11 F(1) = P(X < 1 ) = P(X = 2) + P(X = 1)

= 0.1 + 0.1 = 0.20 F(0) = P(X < 0 ) = P(X=2) + P(X=1)+ P(X = 0)

= 0.1 + 0.1 + 0.2 = 0.41 F(1) = P(X < 1 ) = 0.62 F(2) = P(X < 2 ) = 0.93 F(3) = P(X < 3 ) = 1

vdnt F(x) = 0 , x < 2 v = 0.1 , 2 < x < 1 v = 0.2 , 1 < x < 0 v = 0.4 , 0 < x < 1 v = 0.6 , 1 < x < 2 v = 0.9 , 2 < x < 3 v = 1 , x > 3 v.

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 4X vw rkth- kh fjf rhvw rkth- kh fjf rhvw rkth- kh fjf rhvw rkth- kh fjf rhvw rkth- kh fjf rh

tUkhW.tUkhW.tUkhW.tUkhW.tUkhW.

X : 0 1 2 3p(x) :

61

21

103

301

(i) P(X < 1) (ii) P(X < 2) (iii) P(0< X < 2),tiw fhf.,tiw fhf.,tiw fhf.,tiw fhf.,tiw fhf.

78

:

(i) P(X < 1)= P(X = 0) + P(X = 1) = p(0) + p(1)= 6

1 + 2

1 = 3

2

(ii) P(X < 2)= P(X = 0) + P(X = 1) + P(X = 2)

= 61

+ 21

+ 103

= 3029

bjhU Kiw

P(X < 2)= 1 P(X > 2)

= 1 P(X = 3) = 1 301

= 3029

(iii) P(0 < X < 2) = P(X = 1) = 21

8.1.4 bjhl rkth- khbjhl rkth- khbjhl rkth- khbjhl rkth- khbjhl rkth- khbjhlahd (continuous) kfis VF/ mjhtJ

tiuaWfgl X ,ilbtYs vyh kfisbgwty rkth- khna/ bjhl rkth kh vdgL.

cjhuzkhf,

(i) kiHeh bgh kiH ms(ii) jegf cauf (iii) jegf vilf8.1.5 fjf ml rhfjf ml rhfjf ml rhfjf ml rhfjf ml rh (Probability density

function)rh f MdJ X vw bjhl rkth- kh

fjf rh myJ fjf ml rhghf (p.d.f),UfntLkhdh fl gjidfis iwbr-antL.

(i) f(x) > 0 (x - vyh kfSF)

(ii)

)(xf dx = 1

79

FFFFF :

(i) rkth- kh X, (a, b) vw ,ilbt ,Ugjfhdfjf P(a < X < b) =

b

a

xf )( dx.

(ii) P(X = a) = a

a

xf )( dx = 0

(iii) P(a < X < b) = P(a < X < b) = P(a < X < b) = P(a < X < b)8.1.6 bjhl gut rhbjhl gut rhbjhl gut rhbjhl gut rhbjhl gut rh

X vgJ/ f(x) vw fjf ml rhig cilaxU bjhl rkth- kh v FX(x) = P(X < x)

=

x

tf )( dt

vw rh X - gut rh myJ F gut rh vdmiHfgLwJ.

gfgfgfgfgf :F gut rh fl gfis bgWsJ.

(i)xtL F(x) = 0 mjhtJ F() = 0

(ii)xtL F(x) = 1 mjhtJ F() = 1

(iii) bjhl rkth- kh X - F gut rh F kWfjf rh f v F tifljf f

f(x) = dxd F(x)

vLJfhLvLJfhLvLJfhLvLJfhLvLJfhL 5bjhl rkth- kh bjhl rkth- kh bjhl rkth- kh bjhl rkth- kh bjhl rkth- kh X - fjf ml fjf ml fjf ml fjf ml fjf ml

rhrhrhrhrh

f(x) =

116

9.1 TbwLj kW iHf tiffTbwLj kW iHf tiffTbwLj kW iHf tiffTbwLj kW iHf tiffTbwLj kW iHf tiff(Sampling and types of errors)

TbwLj e mwhl th gagLjgLwJ.rikF bghGJ xU y nrhiw gj ghJ mitf ewhfrikfgLsdth vd xUt nrhgJ kW jhatif xid thf U a

Documents

Std12-BM-TM-2