Chapter 1Chapter 11 Chapter 1Chapter 12 (Exact) Chapter 1Chapter 13 Euler's Method Chapter 1Chapter 14 ()(Differential EquationD.E.)Chapter 1Chapter 15 1. 2. 3. 4. 5. Chapter 1Chapter 16 1. (Variable)(D.E.)(Order) Chapter 1Chapter 17 2. (1) (Ordinary Differential EquationO.D.E.): (Independent Variable)[1](2) (Partial Differential EquationP.D.E.): [2] (3) (Quasi Linear) [5] Chapter 1Chapter 18 (1) (Degree) : [4] (2)(Linear) (Dependent Variable)[1][2](P.D.E.81) (3) (Quasi Linear) [5](4)(Nonlinear) (2)(3) 3.[Note] Chapter 1Chapter 19 (1)(General SolutionG.S.) (2)(Particular SolutionP.S.) (Initial ConditionI.C.)(BoundaryConditionB.C.)() 4. Chapter 1Chapter 110 (1) ) (x f y =A. xe c y =B.c x x y + + =2 4 (2) c y x F = ) , (A.c y xy x = + 2) cos(B.c e x y xy= + + 23 2 5. Chapter 1Chapter 111 0 ) , , ( ='y y x F F x y y'(Arbitrary Constant) cxe c y = 0 = 'y y cxe c y = 0 = 'y y xe y 2 = xe y 3 = 0 = 'y y () 0 ) ( ) ( = + dy y N dx x M (21) Chapter 1Chapter 112 c } }} } }= ++ = +c dy y N dx x Mc dx dy y N dx x M) ( ) (0 ) ( ) ( x y [Note]) (x M x ) ( y N y Chapter 1Chapter 113 0 ) , ( ) , ( = + dy y x N dx y x M (22) ) ( y , x M ) y , x N( (Homogeneous) ) , ( y x M ) , ( y x N (22) ) () , () , (xyfy x Ny x Mdxdy= = (23) ) (xyf xy Chapter 1Chapter 114 (1) xu y uxy= = , ) 3 2 ( ) ( '+ ='+ = u x u ydxdux udxdy (2) u u fdxdux u fdxdux u = = + ) ( , ) (dxxduu u f1) (1= () uxy= (22) D.E. x u D.E. (3) xyu = D.E. Chapter 1Chapter 115 (Exact) 0 ) , ( ) , ( = + dy y x N dx y x M (24) xy x Nyy x Mcc=cc ) , ( ) , ( (25) () Chapter 1Chapter 116 (1) c y x F = ) , ( (24) ) (c d dyyFdxxFdF =cc+cc=0 =cc+ccdyyFdxxF (a) (2) (24)(a) cc= cc=yFNxFMc cc=cccc=ccc cc=cccc=ccy xFyFx xNx yFxFy yM22) () ( MN ) , ( y x F y xFx yFc cc=c cc2 2 xNyMcc=cc() 0 ) , ( ) , ( = + dy y x N dx y x M [] Chapter 1Chapter 117 [ ) , ( y x M x ] (1) xFMcc= x M F c = c }+ = ) ( ) , ( ) , ( y A dx y x M y x F (b) x y y ) ( y A (2) (b) y N y A dx y x My yF= ' +cc=cc}) ( ] ) , ( [ ) ( y A' ) ( y A' y ) ( y A (b)(24) c y A dx y x M y x F = + = }) ( ) , ( ) , (Chapter 1Chapter 118 [ ) , ( y x N y ] (1) yFNcc= y N F c = c }+ = ) ( ) , ( ) , ( x B dy y x N y x F (c) y x x ) (x B (2) (c) x M x B dy y x Nx xF= ' +cc=cc}) ( ] ) , ( [ ) (x B' ) (x B' x ) (x B (c)(24) c x B dy y x N y x F = + = }) ( ) , ( ) , (Chapter 1Chapter 119 0 ) , ( ) , ( = + dy y x N dx y x M ) , ( y x I ) , ( y x I (Integrating Factor I.F.) 0 ) , ( ) , ( = + dy y x N dx y x M ) ( y , x I ) , ( y x I 0 ) , ( ) , ( ) , ( = + dy y x N y x I dx y x M (26) Chapter 1Chapter 120 ) , ( y x I 0 = + Ndy Mdx 0 ) ( ) ( = + dy IN dx IM xINyIMcc=cc ) ( ) ( xINxNIyIMyMIcc+cc=cc+cc(d) Chapter 1Chapter 121 (1) ) ( ) , ( x I y x I = x 0 ) ( =ccx Iy(d) dxdINxNIyMI +cc=cc( ) (x I x ) dxdINxNyMI =cccc) ( NxNyMdxdIIcccc=1 x ) (x f ) (x fNxNyM=cccc D.E. ) (x I}=dx x fe x I) () ( NxNyMx fcccc= ) ( (27) Chapter 1Chapter 122 (2) ) ( ) , ( y I y x I = y 0 ) ( =ccy Ix(d) xNIdydIMyMIcc= +cc( ) ( y I y ) ) (yMxNIdydIMcccc=MyMxNdydIIcccc=1 y ) ( y g ) ( y gMyMxN=cccc D.E. ) ( y I}=dy y ge y I) () ( MyMxNy gcccc= ) ( (28) Chapter 1Chapter 123 (3)(1)(2)b ay x y x I = ) , ( a b 22 Chapter 1Chapter 124 ) ( ) ( x q y x p y = +' (29) ] ) ( [) ( ) (}+ =} }c dx x q e e ydx x p dx x p (210) } p( x) dxe Chapter 1Chapter 125 [](1) 0 ) ( = x q 0 ) ( = + y x pdxdy 0 ) ( = + dx x pydy () (2) 0 ) ( = x q ) ( ) ( x q y x p y = + '}dx x pe) ([] ) ( ) () ( ) ( ) (x q e y x p e y edx x p dx x p dx x p} } }= + ' ) ( ] [) ( ) (x q e y edx x p dx x p} }= ' Chapter 1Chapter 126 c dx x q e y edx x p dx x p+ = }} }) () ( ) ( (211) ] ) ( [) ( ) (c dx x q e e ydx x p dx x p+} }=} (210) [Note] (1) ) ( ) ( x q y x p y = +'y' 1 (2)) (x p ) (x q Chapter 1Chapter 127 []}=dx x pe x I) () ( (1)) ( ) ( x q y x p y = +' ) (x I D.E. Iq Ipy y I = +' 0 ) ( = + q py IdxdyI0 ) ( = + Idy dx q py I Chapter 1Chapter 128 (Bernoulli) ay x q y x p y ) ( ) ( = + ' (212) a () (1)0 = a ) ( ) ( x q y x p y = + ' D.E. (2)1 = a y x q y x p y ) ( ) ( = + ' 0 ] ) ( ) ( [ = + ' y x q x p y D.E. (3)= a ay u=1 D.E. D.E. Chapter 1Chapter 129 [] (3) A. ay x q y x p y ) ( ) ( = + ' ay) ( ) (1x q y x p y ya a= + ' (e) B.ay u=1 dxdyy adxdua = ) 1 ( dxduay ya= '11 (e) ) ( ) (11x q u x pdxdua= + ) 1 ( a ) ( ) 1 ( ) ( ) 1 ( x q a u x p adxdu = + ) ( ) ( ) 1 ( x P x p a = ) ( ) ( ) 1 ( x Q x q a = D.E. ) ( ) ( x Q u x P u = + ' D.E. 26 D.E. C.ay u=1 Chapter 1Chapter 130 1. a2.a3.a Chapter 1Chapter 131 0x x = (1) 0x x < 1y0x x > 2y D.E. (2) 0x x = 2 1y y = 1. x,y(yx)2.3. ) ( ) ( ) ( ) ( x f y g x p y y g = + ' ' ) ( y g = u y y g ' ' = ' ) ( u D.E. ) ( ) ( x f x p = + ' u u D.E. Chapter 1Chapter 132 0 ) , , ( = ' y y x F 0 ) , , ( = p y x F pdxdyy = = ' ()[ 6] [ 6](1)ydxdyxdxdy= +2) ((2)y xp p = +2(dxdyp = ) n 0 ) , ( ) , ( ) , ( ) , (1122 1= + + + + +n nnp p y x f p y x f p y x f y x f Chapter 1Chapter 133 1.(Clairaut) (Lagrange) ) (dxdyfdxdyx y + = (213) dxdyp = ) ( p f p x y + = (214) (Clairaut) ) (c f x c y + = (215) Chapter 1Chapter 134 [] (1) (214) x dxdpp fdxdpx pdxdy) ( ' + + =dxdpp f x p p ] ) ( [ ' + + =0 ] ) ( [ = ' + p f xdxdp 0 =dxdp0 ) ( = ' + p f x(2) 0 =dxdp c p = (214) ) (c f x c y + = (f) D.E. p c (3) 0 ) ( = ' + p f x ) ( p f x ' =) ( )] ( [ ) ( p f p p f p f xp y + ' = + = + ' =' =) ( ) () (p f p f p yp f x (g) Chapter 1Chapter 135 2.(Lagrange) ) ( ) (dxdygdxdyf x y + = (216) dxdyp = ) ( ) ( p g p f x y + = (217) ) ( ) ( ) , () , (p g p f c p h yc p h x+ == (218) Chapter 1Chapter 136 [] (1) (217) x dxdpp gdxdpp f x p fdxdy) ( ) ( ) ( ' + ' + = ) ( pdxdy=dxdpp g p f x p f p ] ) ( ) ( [ ) ( ' + ' = ) ( ) () (p g p f xp f pdxdp' + ' =(2) xp D.E. ) () () () (p f pp gxp f pp fdpdx'=' D.E. ) , ( c p h x = (h) (3) (h)(217) Lagrange D.E. =+ =) , () ( ) ( ) , (c p h xp g p f c p h y Chapter 1Chapter 137 1. (Kirchhoff 's Voltage LawKVL) 2. (Kirchhoff 's Current LawKCL) Chapter 1Chapter 138 (Closed Loop)0() 1. (Kirchhoff 's Voltage LawKVL) Chapter 1Chapter 139

2. (Kirchhoff 's Current LawKCL) (Node)0() 1. RCRL(V) 2. RL(I) 3.RCRL(v(t))Chapter 1Chapter 140 22RC RVCSW+0 = t) (t iChapter 1Chapter 141 23 RC) ( A i01 . 0101 . 0e201 . 0e0 1 2 34 5(sec) tChapter 1Chapter 142

24RL RVLSW+0 = t) (t iChapter 1Chapter 143

25RL ) (t iRV00tt tChapter 1Chapter 144

26RL R) (A ILSW0 = tRiLiChapter 1Chapter 145 ycx y =2 2 2k y x = +x0=k2=k3=k27Chapter 1Chapter 146 1.2.3.Chapter 1Chapter 147 1.(Translational Motion) (Displacement ) (t x )(Velocity ) (t u )(Acceleration ) (t a) t ( F)(M)( a) a M F = (219) 28 M Chapter 1Chapter 148 28 M) (t f) (t x 22) ( ) () ( ) (dtt x dMdtt dM t Ma t f = = =u (220) ) (t a ) (t u ) (t x ) (t Chapter 1Chapter 149 2. [] (1) 32.18 180 (2) 0 = t 3 0100 200 e y = 100 100 200 = = y 100 () Chapter 1Chapter 150 3. Chapter 1Chapter 151 Eulers Method 0 ) , , ( ='y y x F (22 ) ) , ( y x f y =' ) , ( y x f y =' 0 0) ( y x y = (221) Chapter 1Chapter 152 ) (0 0x y y = (221)y c + = =0 1x x x 1y c 20 2+ = = x x x 2y (c 0.2 , 0.1 , 0.01 0.001) Chapter 1Chapter 153 (Taylor Series) + +' '+ ' + = +3) 3 (2! 3) (! 2) () ( ) ( ) ( c c c cx y x yx y x y x y (222) (221) ) , ( y x f y = 'xff ycc= ' = ' 'yyf'cc+ (222) +' '+'+ + = +3 26 2) ( ) ( c c c cf ff x y x y (223) ff 'f ' ' ) ) ( , ( x y x c (223)2c 3c ) ( c + x y c c ) , ( ) ( ) ( y x f x y x y + + Chapter 1Chapter 154 c ) , (0 0 0 1y x f y y + = ) ( ) (0 1c + = x y x y c ) , (1 1 1 2y x f y y + = ) 2 ( ) (0 2c + = x y x y , 2 , 1 , 0 ) , (1= + =+n y x f y yn n n nc (224) (Euler's Method) ) ( x y 0x x = 29 Chapter 1Chapter 155

29y) , (0 0y x0y1y2y2x1x0xc cOx) , (1 1y x f =) , (0 0y x f =Chapter 1Chapter 156 (First Order Method)(223)c (223)c c (223)2c 2c Chapter 1Chapter 157 21(D.E.) Chapter 1Chapter 158 22 0 ) ( ) ( = + dy y N dx x M c 23 D.E. ) () , () , (xyfy x Ny x Mdxdy= = uxy= dxxduu u f1) (1= xyu = 24(Exact) D.E. 0 ) , ( ) , ( = + dy y x N dx y x MxNyMcc=cc ) , ( y x F }= + = c y A dx y x M y x F ) ( ) , ( ) , ( }= + = c x B dy y x N y x F ) ( ) , ( ) , ( 21 Chapter 1Chapter 159 21 () 25 D.E. 0 ) , ( ) , ( = + dy N y x I dx M y x I ) , ( y x I (1) }=dx x fe x I) () (NxNyMx fcccc= ) ((2) }=dy y ge y I) () (MyMxNy gcccc= ) ((3)b ay x y x I = ) , ( a b 26 D.E ) ( ) ( x q y x p y = + ' ((

+} }=}c dx x q e e ydx x p dx x p) () ( ) ( 27 D.E. D.E. ay x q y x p y ) ( ) ( = + ' (1)0 = a ) ( ) ( x q y x p y = + '( D.E.) (2)1 = a 0 ] ) ( ) ( [ = + ' y x q x p y( D.E.) (3)= a ay u=1 D.E. D.E.