Equazioni differenziali - Esercizi risolti.pdf

8/9/2019 Equazioni differenziali - Esercizi risolti.pdf

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Equazioni differenzialiEsercizi risolti

Alessandra Picchiotti

April 25, 2009


2/25

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Alle seguenti condizioni:

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Non commerciale . Non puoi usare questopera per ni commerciali.

Condividi allo stesso modo . Se alteri o trasformi questopera, o sela usi per crearne unaltra, puoi distribuire lopera risultante solo conuna licenza identica o equivalente a questa.

Ogni volta che usi o distribuisci questopera, devi farlo secondo i terminidi questa licenza, che va comunicata con chiarezza. In ogni caso, puoi con-cordare col titolare dei diritti utilizzi di questopera non consentiti da questalicenza. Questa licenza lascia impregiudicati i diritti morali.

1


3/25

cotn( rlRx$\Ri

I

t)^ d.h=

q-NWar.\umt-RsNfrd

f eQet\o q\S\b _

d.(r-t=L-L \\tr,l-nt,

rLe :\.

-L k,- - ib\t = -ib

bL{=)"\t "-iF, L d*..LT i dtrd,b_Jr"_\=- o \."Q, * L \,.-FJ r L J"r"

F{ QQ*Kvo

[$ttr:e. (S,.>:olecsf e,IFterc)=.{ eF

= \ ub- L#\"=. $- dtr*tL'\.i\"

iF,uG)- f. L."k g dr_ Jb*il

cJ el +| uts) - W

+r):c,

t,,,itc:=* r'tr) +(r.+a)

G

NLr-)\:-{ ur :--{a?e- e., r-o

r.Iie.*-dh

=f

a.n")dr=Jqrs*r\i dr=

*\tUt-{l=f;.

L gz- 12q

*c*"-ft

- +L.r '-\r=A**,.-

.') =; G'+.";*s

: f$\:-*,'[=t.urLt): e-toe. L.

2eoa* d


4/25

I , pitc) : -A rrUg)I 3t,

u=)-\:

z dh,=. -,t

l-o*r{F,e,*dh=?\"

dte) = &b" utelN\q=\" -' F - -?N'

t(t>t"- todL:fo-*=A*f=*:a,'.r(e) e,-3bL. _ " "11

eb e.*d,J'=I

ts qoN$tc)=t "'n$ fV)=lt'uo)- pff-. $cc>dtr

(s\RI |\ASe&etr>.

11"*\:I"q =3*-,t-i"*t"

@ rd,trt*l Jqtgl =G-:.) '-" ,t\$ Ct.) = - 11.15)t (b _^),

-?**1

fr- Oo+ot e+bI-b

2cF\ s


5/25

aESRq(

bo,t >o

,ulcel- r\te1-\ .rr"Lr.)-a1,1riltk)

-[*(.]"

ncc)=-L\tfo,.t,)

'SsgsroqriJLt.)p bn?a*

[.^"tt;J--;e

=L",o[.

=-Clnr

=-Lg

oL-L

z)0 *f tt,)

r-tt)[tto:Jt

i (t )

fi$cc;= -3[*a.X+^l(.,)- -3 rrrelL\rs1-+

.tfcel=t cre) _s3l5

nu'tt1 = - r(t) +3t&

fiLel=\t*a r+ a r , dt*dl - -\cek"

B(- ) = Q t ^n, -3r--r" 4'ry-.e *{,. at6dh=} ,. f

= ,uf-?,ote)"**"-_,cf *g .J =*,-o" _3 3.l

^\tb)t"rr-i=^*d%)1.*+= eLc .l_l-\*'' .*

tc) \ Lut'{"t U\ to) =1

-,t te1 = Atr(t)t+,ttte; = _\t) _r

Nfr,tel = ,1 _.rxs _[uos(c) = t -+.\.,ti

do- 6"g=-*" l"'tlce) = s-\ L*_e.*o_\Atr) =[}te)J_.= ed;. -uo

o

qcs\ 4


6/25

ut 6o1 a

utt) = *,b Le_o,.\of.\tte) = q.sLc-e,**dt+ qtrer)fe_*rll i.-ft.'IL= *

ttc-s'+a.J+

"-"L...

=*o,]*--rtq = d (. _4nq\-te+t ) = i+el [o)=r, + lz o=6 c=+

I\l) : *"t (t _*.)-t

,bt .tr.(c)-ts,tl te;: t^rt ll\ gulh. A>o

C r.t (e)= d..t.rr ) _[_rrl Jqx,*rll (b) = .t-o u [b)_ &-bFc.l] *\*^o

,.1tel1ucu5{=_t_L^rtt)fl+ str,.b^)Lk)= Ert*)J--s te1 =-3^\'te) utqlf-4

*^t

(b) : ,^c-{r: Lr) *83 &.\.,.h

dtc) - -b a-{ rtF) +g gr r


7/25


8/25

e,c . .rtsgrtt\aib* =->4{--1l -g.* +r&Fd = -te= uof**+1

KlLk): gb ( c, -c-


9/25

)

O {ur\:\1f tt rg.)

;Ztc-tr ),"Ct:J2bc'ri")

c$\P\SNF|$sficr":\

- 2tzt +s.)utt) 45 r +)--_--j \ _s,Lq *N

s') n t''r,){-urrr=$Aitt)_ o +' ScSLsi rro o= -4 Ltl-r1ha_,rltzt\3) h _"d \+2)

k\2 + '** + p-b\+t\ 1t,\z=o

,1" h3+z\a t) * thi rs\ +z,-or,\o

ftr*eh+t:o [(h+r)._.,Ih,z+rh+a=,, 1 (u*.,tLrz)=o ) [:-o

f,ttt): - {

$ erb)

{&) : a [k)[- o.tr-,1 dott). [itt.)l_ br*)

So*ilu:6.0

\ = a' ht,*o3)+{ &*rlt a(kvr)=*[*g*.t *4{*aev^,,='*+a[bra)

tL)=ton N.h.=at,uob$) = ['o

Jot*o> \b-_\lr*e** oqd"l*).\:oab-**t:$#*)ir.NQs{\Qc)..g rt*):..gCv)=2q?si

f,(b)=eA:-& e

&(e>- *3 (c+e* ) aeq B


10/25

_zbe

+--^( - E e-u*

CNTG\$\)tSERca)) SFRisr).N\ utkJ = - {.

l 1l."t la k\l[t t=l *1"r.rLcl"*futuf="*A

\) ,t t=1- tnts-a

nnth-t. t, ?.-ffo t . ff ret

nqi' -"q-..f*ff*Jl*-ZCvr*t)t\'

+ [nnt) 4)no*.L =o

l ul't-eu = *, ,

#k +hn -a) a +?r=o*-=

\2: nn+2, t (vnz-qvn+


11/25

C,$q(EpEAN)eF?rra

L{r$.r:rE\}k

8 *r-(6.s'r,)-t*. 3 u.tc.) *\6_tr1J 's

qisI ri,t


12/25

gc"-:


13/25

@' xrLe)- - ..*E*orl

t) cos.rtr)l-o

^,"t*):S+[t ccu e\Nu(b'r\.) P ^,f*l c\b\ -;--- S, ).rc) = \ -t d_h"Yceto{^*q db'

Jt-* \Le) - frh"rr.rt\o) -_h +\ol-'r'alL;.1'ts) = --b t ho +*nrr. ,rltt

cs\q(e\r NnQ.\FEIl

"t (s) * orr,*to*r( *Ji,+t+*.qr**r\t ))

[q * .t):-rcd,$r\)L

^t(o):o

r ) ccs \(s)=o :,.U-c) - +h(t (n$i,\*,1.q

tAtt) -

"kl.rI\,(c,)l =,-tr|trtq) \tp

ill:cs.6*-+"*lr +hrs.otu.r)

.s\-( 4fu +\ $.(fio)) =o

* q * &u.,.,"t,to;,=oet. 13


14/25

r(t) = asu(s$xl (_ * )

=e($;

cqfe|{qqlsR"S6

4

6.- Q{,O6 er\re.Q,

Atr\ . *\^tt*.)'-%5 .lrr_r..)

^\ U\")= Ffr +{rskytrrt) _-*$tl

t$t) _ to(t)

n .\ r , r,11r$ = ab.Lt)+,t^$tb)

^\-4./+F = g/+ ,t.t

ct=G,z,;)-o lr=;-* (t*.1

A"'\

=(#*,)-i^\Lt)

J(

\)

\.)

t=4j

Nst* lL

[ $^f$.',,\: t *

=$qfr{*,#,$Lq).aiq,

su,ul*v\o*-,rJf - $rttlr*_$g\"F..t

,=& i' {r. ,

L-t'o

{tt}" $.'u..,.

s ) ^re)t

2"".\k


15/25


16/25

q\REl.\FlSA. . .z

Q&IL\'S* = A( (ih-=4(:tan-r): 6 ( zs+o roh r,oh r ) ==6(asb1r"u\q) == 6( e h. a)z )-* s b e-R-

o h=a,5

\=e

4^* q{8"+ .. i \ n *-l.. *: . \ - ') z( \ B _J=O

+*-{*+\:s

(@at'=*a *I

,,-''(-t). *,-,*.+c.*t**

" *.o h-p?

**, *=A(sh-r) r 4uh "1

//

E\

\

\r

,A_f 'u,,= et-It u"=.tet=

r 4t '\ ,tlt -. e, -t ^ue*9o-3*P,Irt\)- ci

"{* a c:. *(**f

fr2

eo+ \-


17/25

csb\ssssea\ |&,l :. Arl tf.)

x}-.r:o

*C* -q).=o

*tG):= q{

o.

'f, atg +ca e, '

-{c.errso oRn^t: -A) {r{w,l})tl.-q .r.\\r)=k+ ax.\,(e

rr\ ($ = e u Lt) t"rL+ee_s


18/25

@

@

(r-f Ltl =' -'Lh.Lt) -s e,\+*{

a- d.*=t {-o= e,=

1=e'u^ = e-\-

t $::d"*-P = 4.*r v'c*r{ \tuc\L\Lter

A*(')=' **(k^ s-\,.q= i* hz, ** f L) *2 =4

** }r{.

{^l= e\'

1*^*

q: ( h-{t*u\ +h, cssc\

-{ w-tx

c{irvor$\ \ssi.*"

csh\,t(H3\$\egqta*:Al

- ' a tcl *[8:olr- e-t \*-__--q*eh-=

It* *t*\\** *n* \

d -1"* -) \2; - [a1tre*Xr,_X\

d"f eu&-d.*t{ *^

wt*

\sJ_\ts---l ,)

-4 -'s e3x^r,t

J.\ rtu.):

\*n d= \\** -"-" \

- s**l.,(et* e3 d ): -a K\rl-d sv{ -* *'s

G .rt r.r * xt{"L)'it,r r.\r.=

^$o) - uLr) +(b,rr) e-o-

* q qsDfi'\3te${dr-*:*% =4-

*=t+

(,l.to = e,=J

l*o=q'&

* Ae,ia:h*,lfe- csr_h*\rpi

{* '*s L|A\=l e- \=-u.b-d*-o-c-e '= *

[*l :d* \ Zo"q 1i


19/25

. =*1r-i\=q,=\go \=

cs{r,sfF^}Qercedo,i

tr

* U Lt+ ") as-= Q--- t+a2t

(-. +\"..*,* =_re.[**\:=+ t'{+"' nk-t*=*\ (tr)f"a1=*f"*"od.k*fo-2\. , , F ')o ^L* \o"-*r* l=ouF";.uqr I La

= * I b"**

{ eo + g,"*_ :{ { = _{ foee+g .* * -E- 4 4 - -B

rLt){ t *-i ") "" bt \a


20/25

: t?. "" lt- \ ",ro oooAt'-z"tf"=

O r.'.e,1Gt-z')-- u11*>\t u tdl=o

I ot$-Lr):\ _tdr __h

$oa.t

\r{")-\e"-ar f ot--nb

\o

u.* *o- -at e "-- \" : ' "= d. = oes .g{${.b: bu*gs.r=o\CkI =.* =, gG) =et

co$.sltlNereedr

= "t"et-2ts- , _r\"

Ar.(b):

tb g\ t, - -f er\-e\ *s{a*s

\a-"t**[\"

n\'j'*

"**]-

$3|,:*'I;i.."=z+ eF-zt qb+ z *'\" * k2 eb-at=" _rrE+/* r_/_t, tb(t *.)\o

zr.rsg)-o L"rs*(c-";1 u (r) =o: *l L. + (*+)l=s, Q='e '

u\G)= rs uLtt-rk-t+z)sr{&):I. "

cJ.k--l*.9 :t2

$tr=)-\",**o)it"'dtr = \t * L=.L*0o ,J"

. 'Ll* fc*n-f '-^\" '

--.*fo'..+"\it'*

^oo\ 2


21/25

Co$ffb\^feweda-,e .rr.\t) =.\\ , r

^\\t)

)- (l: --,1") d tr-l *.(fJ -{,i ) ^r(\)

- A3*+') -u..tc,l tdq":e i) _ri"Lt:c:

- qq cas

e*r-C-i) ,*r(g-6. ) -.,

3*rC'. -i .* +(^t 6-1.e.

+t1: t \


22/25

e t'

c\: &| 69.aq,aavrszk,)cQca\FrF$

a6\

c^t = *" "*" d- (t * e* r\l..J:,) - d* ( t-*eR_&u *16

f.Dsco =.\"-l d" Gte*r;,ar:r-)d..l"-4*.-"ltr"f-.gt*,,

o 6 v \\ \v -o

J"-Z--

v \r\

.JoD\ .r -bt, N\-o? * '\Kuzrtt i'hk*,31-*"o*

\.. *-.**".,,2ru\-3se [ * *" l\'o4 *"*o \'* *3t'-A \ *b 1 \. (3 * \o- 3 J" * rc$s.c{fF.c\\=

: ". '*(**)"6 n*" j"FlO

-b q3\\'2Jq"*.ZCSS:ti,

(-t)'+ z'

"r3 'K^E:

3sr t,b e t l:***(*-\)- )-#to-{)+-f+q t 4g v- i/ - 'B tJ \--3/


23/25

=G **1ss*r*+a.ciur.->l e,+ar*a\ll*sax) - t x,^t^=

:*

x*rse9*dr.l'x*o-".,,.x,"lffi (_h,*_"**l::3 "+ *tT( t**-ar*",rr,f_=-h *r?o(g,**^ + ac,rs,:c)o*g

u(r) = qb co* #=..u

aocr 23


24/25

ts \ '\+ur&)" I l-Ntr)

I"l+rr.tc,)1l- l :l.,t-.t.rct)

\*r *|*zr,: q

,'f'' \qtf,, coK, +C- g -Srrr\ccg.,t-c.--t

,[tALt) =, qb-\*'t5*" u- ou\r

e-xr$-


25/25

^lLlc) = ,&, Gts4:j') -rL

Sgconruo orq.s r.F tc>Eg _

,.f tC:-=

.\t,,u.)u\ ,. *(r_.tltrtt)- e$.xsr:ao* (

n &ra).ex\ \\.s.z _(t+a*)**G_i\

o-tt*) a +e._l):o

A: (l-"t*l-.\L * = ,t qi. +sr{;sr-r*

.E\qri

0

{s"t_4tX={s

?.^= 4*: r'--- ---- 'al-2 -\

\a

{sr4rr)

| ,\.\z(t)

b e^h

: {k'*i;

,-q-'c + La *tttl )ute) : c--{

G $ u*tr)= , t,* , . \ ,i$l\i , *.i) \-}.r) + k:\a -:5)

.tr[t) =c.*r.HqR_

'-tq' cex*s$Srcsi

4?2 - +(sE*r).* +Q"h-r)=g

-- ,+(zsf*r'-\()$ + +twh.*5; S(esh, ' t I ric \. .7lo *s ):.r 7'

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Equazioni differenziali - Esercizi risolti.pdf