1.Chapter1 ODE.pdf

7/24/2019 1.Chapter1 ODE.pdf

1/44

1

1.1

(Differential Equations)

,

1.2

1

( )dy

y f xdx

= = (1.1)

y x (dependentVariable) y (independent Variable)

(ordinary differential Equation: ODE)

2 0y xy + = (1.2)2

2

2 0

d yk y

dx+ = (1.3)

11

-

-

x dyy

y dx= (1.4)

32

2 2 0

d y dy

dx dx

+ = (1.5)


2/44

4

(partial differential

Equation : PDE) 22

u uk

t x

=

(1.6)

f fx y nf

x y

+ =

(1.7)

2

(1)

(linear ODE) ()

(2) (nonlinear ODE)(nonlinear term)

1.1

.)2

2 4 2 sin( )

d y dyy x

dx dx

+ + =

.) 4 2 sin( )y yy y x + + = .) cos( ) 0y y + =

.)

2

2 , , 2

d y d y

ydx dx sin( )

.) yy

.) cos( )y


3/44

5

2 (1) (homogeneous ODE)

(2)

(nonhomogeneous ODE)

1.2

.)2

2 4 2 0

d y dyx y

dx dx

+ + =

.) 4 2 sin( )y xy y x + + = .)

.) x sin( )x

(order) (degree)

1.3

.) 4 2 sin( )y xy y x + + = .) 2 2 2 2 2( ) 2 ( ) 0y x yy x y x y + + =

.) 2 1 y .) 3 2 2( )y


4/44

6

(solution)

2y xy= 2xy Ce=

C(arbitrary constant)

1.3

( , )dy

f x ydx

= (1.8)

1.3.1 (Separation Variable)

( ) ( ) 0f x dx g y dy+ = (1.9)

(1.9) f x g y (1.9)

2 2dyy xdx

= 2 2 0y dy xdx =

2

(3 ) cos( )

y dy

y e xdx+ = 2

(3 ) cos( ) 0

y

y e dy x dx+ = (1.9)

( ) ( )f x dx g y dy c+ = (1.10)

c 1.4

22dy

xydx = (1.11)


5/44

7

(1.11) 2

2xdx y dy

= 22 -dx y dy c= +

2 1 cy

= +

21 x cy

=

2

1-yx c

=+

## (1.12)

(1.12) (general solution)

1.5 2( 3)y dx xydy+ = (1.13)

(1.13) 2

03

dx ydy

x y =

+

2 31 03

dxdy

x y

= +

32 1

3-

dxdy c

x y

= = +

2 ln 3ln( 3) ln-y y c+ + =

2 ln 3ln( 3) ln-y x y c= + +

2ln ( 3)y x y c= +

2 3( 3)ye cx y= + ##

1.6

2

cos

3 ydy x

dx y e=

+ (1.14)

(0) 2y = (1.14)

2(3 ) cosyy e dy xdx+ =


6/44

8

3

siny

y e x c+ = + (1.15)

(1.15) (1.14) (0) 2y = 0, 2x y= = c (1.15)

28 e c+ = (1.16)

(1.16) (1.15) 3 2

sin 8

y

y e x e+ = + +

## (1.17) (1.17) (particular solution) c

1.3.2 (Exact Equation)

2 y ( , ) ( , ) 0M x y dx N x y dy+ = (1.18)

(1.18) (standarddifferential form) (1.8)

( , )dy

y f x ydx

= =

(1.3.1) ( , ) ( 1) 0f x y dx dy+ = (1.19)

(1.18) (1.19) ( , )

( , )

dy M x y

dx N x y= (1.20)

(1.18) (1.20) (1.18)

3 2( 2 ) (3 1) 0y x dx xy dy+ + + =


7/44

9

(1.18) dx dy

x y 2 (1.18) ( , ) ( , )x y dx N x y dy+

f f

df dx dyx y

= +

(1.21)

(1.18) (1.21) ( , )f x M x y = ,

( , )f y N x y = ( , ) 0d f x y = 0

f fdx dy

x y

+ =

( , ) ( , )x y dx N x y dy+ f ( , )f x M x y =

( , )f y N x y =

0f f

dx dyx y

+ =

( , ) 0d f x y =

( , ) ( , )x y dx N x y dy+ ( , ) 0d f x y = ( , )f x y c= c 1.) ( , )x y ( , )N x y ,y

N 2.)

1.7 3 2( 2 ) (3 1) 0y x dx xy dy+ + + =


8/44

10

3 2( 3 ) 2 0y dx xy dy xdx dy+ + + =

3 2( ) ( ) 0d xy d x dy+ + =

3 2( ) 0d xy x y+ + =

3 2xy x y c+ + =

3 2( , )f x y xy x y= + + ##1.7

1.1 y

N

y

R

xy ( , ) ( , ) 0M x y dx N x y dy+ = N

y x

=

( , )x y R

1 ( , ) ( , ) 0M x y dx N x y dy+ = ( , )f x y f x= N f y=

y N x 2f

y y x

=

(1.22a)

2N f

x y

=

(1.22b)


9/44

11

(1.22a) (1.22b)

( , ) ( , ) 0M x y dx N x y dy+ = N

y x

=

Ny x

=

f

f x M = f y N = ( , ) ( , ) 0M x y dx N x y dy+ =

f x M = f x y t(dummy variable)

0

( , ) ( , ) ( )x

x

f x y M t y dt c y= + (1.23)

0( , )y R ( )c y y

(1.23) y N ; ( fNy

=

)

0 0

( , )( , ) ( ) ( )

x x

x x

f M t yt y dt c y dt c y

y y y

= + = +

(1.24)

(1.24) ( )c y ( , ) ( , )x y N x y

y x

=

t ( , ) ( , )t y N t yy t

=

(1.24)

0 0

( , ) ( , )( ) ( )

x x

x x

f M t y N t ydt c y dt c y

y y t

= + = +

0( , ) ( , )( )

xxf N t y N t y

dt dt c yy t t

= +

0

( , ) ( )t x

t x

fN t y c y

y

=

=

= +

0( , ) ( , ) ( )f

N x y N x y c yy

= +

(1.25)

(1.25) ( )c y 0( , )N x y

fy

( , )N x y

0

( ) ( , )c y N x y =

( )c y


10/44

12

0

0( ) ( , )

y

y

c y N x t dt = ; 0 0( , )y R

y N x =

0 0

0( , ) ( , ) ( , )

yx

x y

f x y M t y dt N x t dt= + (1.26)

(1.26) ,f x M f y N = =

( , ) ( , )f f

df dx dy M x y dx N x y dyx y

= + = +

f y N =

f N y x g

0 0

0( , ) ( , ) ( , )

yx

x y

g x y M t y dt N x t dt= + (1.27)

( , ) ( , ) 0M x y dx N x y dy+ =

R

0 0( , )y

R

0 0

0( , ) ( , ) ( , )

yx

x y

f x y M t y dt N x t dt k= + = (1.28)

0 0

0( , ) ( , ) ( , )

yx

x y

g x y M t y dt N x t dt c= + = (1.29)

k c

1.8 1.73 2( 2 ) (3 1) 0y x dx xy dy+ + + =

3( , ) 2x y y x= + , 2( , ) 3 1N x y xy= +

23M yy

=

, 23

Ny

x

=

(1.28)


11/44

13

0 0

0( , ) ( , ) ( , )

yx

x y

f x y M t y dt N x t dt k= + =

0 0( , ) (0, 0)x y = 3( , ) 2t y y t = + 2

0( 0, ) 3(0) 1 1N x y y= = + =

(1.28)

( )30 0

( , ) 2 1

yx

f x y y t dt dt k= + + =

( )3 002x y

y t t k+ + =

3 2y x x y k+ + = ##

(1.29)

0 0

0( , ) ( , ) ( , )

yx

x y

g x y M t y dt N x t dt c= + =

0 0( , ) (0, 0)x y = 3( , ) 3N x t xt t= + 0( , 0) 2t y t= = (1.29)

2

0 0( , ) 2 (3 1)

yx

g x y tdt xt dt c= + + =

2 3

0 0( )

x y

t xt t k + + =

2 3xy y c+ + = ##

1.9 33 ( 2) ( 2 ) 0x xy dx x y dy + + =

2( , ) 3 6x y x y x= , 3( , ) 2N x y x y= +

2( , ) 3M x yy

=

, 2

( , )3

N x y

x

=

(1.28)

0 0

0( , ) ( , ) ( , )

yx

x y



12/44

14

0 0( , ) (0, 0)x y = 2( , ) 3 6t y t y t = 0( 0, ) 2N x t t= =

(1.28)( )2

0 0

( , ) 3 6 2

yx

f x y t y t dt tdt k= + =

( )3 2 200

3x y

t y t t k + + =

3 2 23x y x y k+ + = ##

1.10

(2 ) 0y ye dx y xe dy + =

( , ) yx y e= , ( , ) (2 )yN x y y xe= +

( , ) yM x y ey

=

,( , ) yN x y ex

=

(1.28)

0 0

0( , ) ( , ) ( , )

yx

x y


0 0( , ) (0, 0)x y = ( , ) yt y e= 0( 0, ) 2N x t t= = (1.28)

0 0

( , ) ( 2 )

yx

yf x y e dt t dt k= + =

2

0 0

x yye t t k =

2ye x y k = ##

1 22 ( , )f x M x y =


13/44

15

y

( , ) ( , ) ( )

x

f x y M x y dx C y= + (1.30)

( )C y y

(1.30) y ( , )N x y ; ( , ) fN x yy

=

( , )

( ) ( , )x

M x ydx C y N x y

y

+ =

( , )( ) ( , )

x x yC y N x y dx

y

=

( )C y ( )C y y ( )C y (1.30)

( , )f y N x y = y x

( , ) ( , ) ( )

y

f x y N x y dy C x= + (1.31)

( )C x x (1.31) x

( , )x y ; ( , ) fM x yx

=

( , )( ) ( , )

yN x y

dy C x M x yx

+ =

( , )( ) ( , )

yN x y

C x M x y dyx

=

( )C x ( )C x x ( )C x (1.31)

1.11 1.933 ( 2) ( 2 ) 0x xy dx x y dy + + = (1.32)

2


14/44

16

2

( , ) 3 6x y x y x= ,2

( , ) 3N x y x=

2( , ) 3M x y xy

=

, 2

( , )3

N x yx

x

=

(1.32)

( , )f x y C=

2( , ) 3 6

fx y x y x

x

= =

(1.33)

3( , ) 2f

N x y x yy

= = +

(1.34)

( , )f x y (1.33) y 2( , ) ( , ) ( ) (3 6 ) ( )

x x

f x y M x y dx C y x y x dx C y= + = +

3 2( , ) 3 ( )f x y x y x C y= + + (1.35)

(1.35) y 3 ( )f x C y

y = +

(1.36)

(1.36) (1.34) 3 3( ) 2x C y x y+ = +

( ) 2C y y =

( )C y 2

1( )C y y C = +

( )C y (1.33)3 2 2

1( , ) 3f x y x y x y C= + + +

3 2 23x y x y C+ + = ##

( )C y ( )C yy


15/44

17

1.3.3(Integrating Factor)

1.3.2

3 2 22 3 0xy dx x y dy+ =

2xy

2 3 0ydx xdy+ =

2xy

( )2 2 3 0xy ydx xdy+ =

2y

( , ) ( , ) 0M x y dx N x y dy+ =

( , )x y

( , )x y ( , )x y

1.12 2 2( ) 0x y x dx ydy+ = (1.37)

2 2( , ) ( )x y x y x= + , ( , )N x y y=

( , ) 2M x y yy

=

,

( , )0

N x y

x

=

(1.37)

2 2( )xdx ydy d x y = +


16/44

18

(differential) 2 2( )x y+ 2 2

( )y+ 2 21

( )x y+ 2 2

2 2 2 2

1 1( ) 0x y x dx ydy

x y x y+ =

+ +

2 2 2 2(1 ) 0

x ydx dy

x y x y =

+ + (1.38)

(1.38)

2 2( , ) (1 )

xM x y

x y=

+,

2 2( , )

yN x y

x y=

+

(1.38)

( )2

2 2

( , ) 2M x y xy

y y

=

+ ,

( )2

2 2

( , ) 2N x y xy

x x y

=

+

##

1.3.3.1

( , ) ( , ) 0M x y dx N x y dy+ = (1.39)

( , )x y (1.39) ( , )y = , ( , )M x y= ( , )N N x y=

0Mdx Ndy + =

[ ] [ ]N

y x

=


17/44

19

M M N Ny y x x

+ = +

1 N M M N

x y y x

=

(1.40)

( , )y (1.39) ( , )y (1.40)

(1.40) y

0y

=

(1.40)

1

0N M N

x y x

=

1 1d M N

dx N y x

=

(1.41)

(1.41) x

1( ) NP xN y x

=

(1.41) 1

( )d

P xdx

=

( )P x dxe =

y 0x

=

(1.40) 1

0 M N

y y x

=

1 1d M Ndy M y x

=

(1.42)


18/44

20

(1.42) y

1

( ) N

Q yy x

=

(1.42) 1

( )d

Q ydy

=

y ( )Q y d y

e =

1.) 1 N

N y x

x

(1.39) 1

exp M N

dxN y x

=

2.) 1 Ny x

y

(1.30) 1

exp M N

dyM y x

=

1.13 ( ) ( 2 1) 0y x y dx x y dy+ + + = (1.43)

( )y x y= + , ( 2 1)N x y= +

2M x yy

= +

, 1

N

x

=

(1.43)

1 2 11

2 1

M N x y

N y x x y

+ = = +


19/44

21

(1.43) (1)dx x

e e

= =

(1.43) xe= 2( ) ( 2 ) 0x x x x xxe y e y dx xe e y e dy+ + + = (1.44)

(1.44) 2( , ) x xx y xe y e y= + ( , ) 2x x xN x y xe e y e= +

(1.28)

0 0

0( , ) ( , ) ( , )

yx

x y


0 0( , ) ( , )x y a b= 2( , ) t tt y te y e y= +

0( , ) 2a a a

N x a t ae e t e= = +

(1.28) 2( , ) ( ) ( 2 )

yx

t t a a a

a b

f x y te y e y dt ae e t e dt k= + + + =

0, 0a b= = 2

0 0

( , ) ( ) (2 1)

yx

t tf x y te y e y dt t dt k= + + =

2x x xe y e y e y k + = ##

1.13 1

1.14 2(4 3 ) ( 2 ) 0xy y x dx x x y dy+ + + = (1.45)

2(4 3 )xy y x= + , ( 2 )N x x y= +


20/44

22

4 6M x yy

= +

, 2 2

Ny

x

= +

(1.45)

1 2 4 2

( 2 )

M N x y

N y x x x y x

+ = = +

x 2

2dx

xe x

= =

(1.45) 2x= 3 2 2 3 4 3(4 3 ) ( 2 ) 0x y x y x dx x x y dy+ + + = (1.46)

(1.46) 3 2 2 34 3

fx y x y x

x

= = +

(1.47)

4 32f

N x x yy

= = +

(1.48)

( , )f x y (1.47)

( , ) ( , ) ( )x

f x y M x y dx C y= +

3 2 2 3(4 3 ) ( )x

x y x y x dx C y= + + 4

4 3 2( , ) ( )4

xf x y x y x y C y= + + (1.49)

(1.49) y

4 32 ( )f

x x y C yy

= + + (1.50)

(1.50) (1.48)4 3 4 32 ( ) 2x x y C y x x y+ + = +

( ) 0C y =

( )C y 1( )C y C= ( )C y (1.49)

44 3 2

1( , ) 0

4xf x y x y x y C= + + =


21/44

23

4

4 3 2

4

xy x y C+ = ##

1.15 ( 1) ( 3 2) 0y x y dx x x dy+ + + + + = (1.51)

( 1)M y x y= + + , ( 3 2)N x x= + +

2 1M yy

= + +

, 2 3 2

Nx y

x

= + +

(1.51)

1 1 1( 1)

M N x y

y x y x y y

= = + +

y 1

dyy

e y

= =

y= 2 3 2 2 2

( ) ( 3 2 ) 0xy y y dx x y xy xy dy+ + + + + = (1.52)

(1.52) 2 3 2f xy y y

x

= = + +

(1.53)

2 23 2f

N x y xy xyy

= = + +

(1.54)

( , )f x y (1.53) x

( , ) ( , ) ( )x

f x y M x y dx C y= +

2 3 2( ) ( )x

xy y y dx C y= + + + 2 2

3 2( , ) ( )2

x yf x y xy xy C y= + + + (1.55)

(1.55) y 2 23 2 ( )

fy xy xy C y

y

= + + +

(1.56)

(1.56) (1.54)


22/44

24

2 2 2 23 2 ( ) 3 2x y xy xy C y x y xy xy+ + + = + +

( ) 0C y =

( )C y 1( )C y C= ( )C y (1.55)

2 23 2

1( , ) 02

x yf x y xy xy C= + + + =

2 2

3 2

2

x yy xy C+ + = ##

1.14 1.15 2

1.3.4(Linear First-Order Equations)

y y

( ) ( ) ( )dy

A x B x y C xdx

+ =

( )A x

( ) ( )dy

P x y Q xdx

+ = (1.57)

( )( )( )

B xP x

A x= ( )( )

( )

C xQ x

A x=

(1.57)

( , ) ( , ) 0M x y dx N x y dy+ = (1.58)

(1.57)

[ ( ) ( )] 0P x y Q x dx dy + = (1.59)


23/44

25

(1.58) (1.59)

( , ) ( ) ( )x y P x y Q x= , ( , ) 1N x y =

( )M P xy

=

, 0

N

x

=

(1.59) ( ) 0P x =

( )x x

(1.59) ( ) ( )[ ( ) ( )] ( ) 0x P x y Q x dx x dy + = (1.60)

( ) (1.60)

( )[ ( ) ( )] ( ) ( )M

x P x y Q x x P xy

= =

( ) ( )NN x xX

= =

(1.60)

( ) ( ) ( )P x x =

( )( ) ( ) d xx P xdx

=

( )

( )( )

d xP x dx

x

=

ln ( ) ( )x P x dx = ( )( ) P x dxx e =

(1.57) ( )x

( ) ( )( ) ( )

P x dx P x dxdye P x y Q x e

dx

+ =

( ) ( )

( )P x dx P x dxd

ye Q x edx

=


24/44

26

( ) ( )

( )P x dx P x dx

e y Q x e dx C = +

( ) ( ) ( )

( )P x dx P x dx P x dx

y e Q x e dx Ce

= + (1.61)(1.61) (1.57)

( ) ( )

1 ( )P x dx P x dx

y e Q x e dx = ( ) ( )

dyP x y Q x

dx+ =

( )2P x dx

y Ce= ( ) 0dy P x y

dx+ =

( ) ( )dy

P x y Q xdx

+ = 1 2y y y= + 1y

2y (Complementary Function)

1.16 2dy y

xdx x

= (1.62)

(1.62)

( ) ( )dy

P x y Q x

dx

+ = 2( )P x = ( )Q x x=

2

( )

2

1dxP x dxxe e

x

= = =

(1.62) 2

1

x

2 3 2

1 2 1 1dy y d y

x dx x x dx x x

= =

2 ln

yx C

x= +

2 2

lny x x Cx= + ##

1.17 2y xy x = (1.63)

(0) 1y =

(1.63) 2

( )P x = ( )Q x x=


25/44

27

2( ) 2P x dx xdx xe e e = = =

(1.63) 2x

e

2 2

( 2 )x xe y xy xe =

2 2x xye xe =

2 2x xye xe dx C = + 21

2

xe C= +

21

2

xy Ce= + (1.64)

(0) 1y = (1.64)1

12

C= +

3

2C =

21 3

2 2

xy e= + ##

(1.61)

1.)

( )P x dxe 2.) 1.

y 1.3.)

y

1.18 2(1 )( ) 2dy dx xydx+ = (1.65)

(0) 1y = (1.65) 2(1 )x dx+

2

21

1

dy xy

dx x =

+


26/44

28

2

2( )

1P x

x=

+

2

2

2( ) exp exp ln(1 )

1

xx dx x

x

= = + +

( )2 1

ln 1

2

1e

1

x

x

+

= =+

2

1

1 x+

y 2

1

1 x+

2 2

1 1

1 1y

x

= + +

y 1

21

yTan x c

x

= ++

2 1 2(1 ) (1 )y x Tan x c x= + + + 1, 0y x= =

1 0 1c c= + = 2 1 2(1 ) (1 )y x Tan x x

= + + + ##

1.4

(,) (resistor) (inductor)

(capacitor) 1.1


27/44

29

1.1 RLC

1.1

1.1

- t (second),s

E =e =

(volt),

(current)- i (ampere), (charge)- q (coulomb), (resistance)-R (ohm)- (inductance)-L (henry)- (capacitance)- C (farad)-F

(voltage drop)1.)

( )RV Ri t = (1.66)

R i


28/44

30

2.) ( )

L

di tV L dt= (1.67)

L i

3.) 1

( )CV q tC

= (1.68)

C q

( )( ) dq ti tdt

=

( ) ( )dq t i t dt =

0

0

( ) ( )

t

Cq t i t dt V = +

0CV

0t=

0

1( )

t

CV i t dt C

= (1.69)

(Kirchhoffs Voltage Law ; KVL)

1.1

L R CV V V E + + = (1.70)

(1.66) , (1.67)(1.68) (1.70)( ) 1

( ) ( )di t

L Ri t q t Edt C

+ + = (1.71)

(1.71)

( )i t

( )q t


29/44

31

(1.71) t( ) ( ) 1 ( )d di t di t dq t dE

L Rdt dt dt C dt dt

+ + =

2 ( ) ( ) 1( ) 0

d i t di t L R i t

dt dt C + + = (1.72)

(1.72) ( )i t

( ) dqi tdt

= (1.71) ( ) ( ) 1

( )Ld dq t dq t

R q t E

dt dt dt C

+ + =

2

2

( ) ( ) 1( )

d q t dq t L R q t E

dt dt C + + = (1.73)

(1.73) ( )q t 1.1 (1.71)

( ) 1

( ) ( ) ( )di t

L Ri t q t e tdt C

+ + = (1.74)

(1.71) RL

( )( )

di t R E i t

dt L L+ = (1.75)

(1.71) RC 1

( ) ( )Ri t q t EC

+ = (1.76)

( )i t ( )( ) dq ti tdt

=

( ) 1

( )dq t E

q tdt RC R

+ = (1.77)


30/44

32

(1.72) (1.73)

2

1.19 RL (E) V 1.2

1.2 RC (0) 0i = (1.75)

( )( )

di t R V i t

dt L L+ =

(1.75)

( )1

( ) ( )di t V Ri t dt L

=

( ) ( )R V

di t i t dt L R

=

( )

( )

di t Rdt

V Li t

R

=

(1.78)

ln ( ) ln

V Rti t k

R L

=

( )Rt

LV

i t keR

=

( )Rt

LV

i t keR

= (1.79)

; k

k (0) 0i = (1.79)


31/44

33

(0 )

( ) 0R

LV

i t keR

= =

VkR

=

k (1.79)

( ) 1Rt Rt

L LV V V

i t e eR R R

= =

##

1.20 1.2 5 .V 50 . 1 .

t 5, 50V R= = 1L= 1.19

50

15

( ) 1 150

Rt t

LV

i t e eR

= =

( )501

110

te=

501 1( )10 10

ti t e A = ## (1.80)

(1.80) 110

(Steady-state

current) 50110

te (Transient current)

t

1.21 1.2

(0) 0i = (1.75)

( )( )

di t R V i t

dt L L

+ =

(1.81)

(1.81)

( ) R

P tL

= ( ) VQ tL

=

( )( )

( ) P t dt R L dt Rt L

t e e e = = =


32/44

34

(1.81) ( )t ( )

( )Rt L Rt L Rt Ldi t R V

e e i t edt L L

+ =

( ) Rt L Rt Ld V

i t e edt L

=

( ) Rt L Rt LVi t e e dt C L

= +

Rt L Rt LV L Ve C e C L R R

= + = +

( ) Rt LVi t CeR

= + (1.82)

C (0) 0i = (1.82)(0 )

( ) 0R

LV

i t CeR

= = +

VC

R=

C (1.82)

( ) 1

Rt Rt

L L

V V Vi t e eR R R

= = ## (1.83)

1.19

1.22 1.2 sinmV t mV (Amplitude) (0) 0i =

(1.77)

( )( )

di t R V i t

dt L L

+ =

(1.84)

(1.84) sinmV t ( )

( ) sinmVdi t R

i t tdt L L

+ =

( ) ( ) sinmVR

i t i t t L L

+ =

(1.85)


33/44

35

(1.85)

( )i t ( )t ( ) Rp t

L=

( / ) /( ) R L dt Rt Lt e e = =

(1.85) ( )t / / /( ) ( ) sinRt L Rt L Rt L m

Vdi t Re e i t e t

dt L L + =

/ /( ) sinRt L Rt L mVd

e i t e t

dt L

=

/ /( ) sinRt L Rt LmV

e i t e tdt C L

= + (1.86)

(1.86) / sinRt Le tdt (by part)

lnu u

da a adu= 1

cos sinnxdx nxn

=

1sin cosnxdx nxn=

;

( )d uv udv vdu= +

udv uv vdu=

Rt Lu e= , sindv tdt = Rt LR

du e dt L

= ,1

cosv t

=

/ / /1 1sin cos cosRt L Rt L Rt LR

e tdt e t t e dt L

=

/ / /1 1sin cos cosRt L Rt L Rt LR

e tdt e t e tdt L

= + (1.87)

(1.87) / cosRt Le tdt

Rt Lu e= , cosdv tdt =


34/44

36

Rt LR

du e dt L

= ,1

sinv t

=

/ / /1 1cos sin sinRt L Rt L Rt LRe tdt e t e tdt L

= (1.88)

(1.88) (1.87)/ / / /1 1 1 1sin cos sin sinRt L Rt L Rt L Rt L

R Re tdt e t e t e tdt

L L

= +

2/ / /

2 2 2

1 1 1cos sin sinRt L Rt L Rt L

R Re t e t e tdt

L L

= +

2

/ /2 2 2

1 1 1(1 ) sin sin cosRt L Rt LR Re tdt e t t L L

+ =

/

2

/

2

2 2

1 1sin cos

sin1

1

Rt L

Rt L

Re t t

Le tdt

R

L

=

+

(1.89)

(1.89) (1.88)

/2

/

2

2 2

1 1sin cos

( )1

1

Rt L

Rt L m

Re t t

LVe i t C

L R

L

= +

+

2

/

2

2 2

1 1sin cos

( )1

1

Rt Lm

Rt t

LVi t Ce

L R

L

= +

+

(1.90)

( 0) 0i t= =

/

2

2 2

10

01

1

Rt LmV CeL R

L

= +

+

2

2 2

1

1

1

mVCL R

L

=

+

(1.91)


35/44

37

(1.91) (1.90)

2 /

2 2

2 2 2 2

1 1sin cos

( )1 1

1 1

Rt L

m m

R t tLV V e

i tL LR R

L L

= +

+ +

2 2 2( ) sin cos Rt Lm

V L Ri t t t e

R L L

= + +

###

1.23 1.2

3sin 2 .t V 10 . 0.5 . 6 ..)

t.)

( )sin 2A t A (phase angle)

.)

(Amplitude), (frequency)(periode)

.) ( ) 3sin 2 , 10v t t R= = 1L= (1.75) ( )E v t

( ) ( )( )

di t R v t i t

dt L L + = (1.92)

(1.92)( ) 10 3sin 2

( )0.5 0.5

di t t i tdt + =

( )20 ( ) 6 sin 2

di ti t t

dt+ =

( ) 20 ( ) 6 sin 2i t i t t + = (1.93)(1.93)

( ) 20p t = ( )t ( ) 20 20( )

p t dt dt tt e e e = = =

(1.93) ( )t


36/44

38

20 20 20( ) 20 ( ) 6 sin 2t t te i t e i t e t + =

20 20

( ) 6 sin 2t t

e i t e t

= 20 20( ) 6 sin 2t te i t e t

=

20

20

1( ) 6 sin 2t

ti t e tdt C

e= + (1.94)

(1.94) (1.87),(1.88),(1.89) (1.90)

2030 3( ) sin 2 cos 2101 101

ti t t t Ce= +

(1.90) (1.22)

2

/

2

2 2

1 1sin cos

( )1

1

Rt Lm

Rt t

LVi t Ce

L R

L

= +

+

210 /0.5

2

2 2

1 10 1sin 2 cos 2

3 2 0.5 2

0.5 1 1012 0.5

t

t t

Ce

= +

+

20

1 120sin 2 cos 2

4 26

11 400

4

t

t t

Ce

= + +

2030 3( ) sin 2 cos 2101 101

ti t t t Ce = + (1.95)

6 .A (1.95)20(0)30 36 sin 0 cos 0

101 101Ce= +

609

101C=

t20609 30 3( ) sin 2 cos 2

101 101 101

ti t e t t

= + A.

20609

101

te 30 3sin 2 cos 2101 101t t ##


37/44

39

.)

( )sin sin cos cos sin = ( )sin 2A t

( ) ( )sin 2 sin 2 cos cos 2 sinA t A t t = (1.96)

SI 30 3

sin 2 cos 2101 101

SI t t = (1.97)

(1.96) (1.97)

30cos101

A = (1.98)

3sin

101A = (1.99)

11 1tan10 10

Tan = =

2 2

2 2 2 2 30 3 3cos sin101 101 101

A A

+ = + =

3

101

A =

13 1sin(2 ) ;

10101SI t Tan

= = ##

.) (Amplitude) 3101

A =

2 f = ; 12

f

= = 1T

f= = ##

1.24 1.2 RL R L

0I t

(1.75)( )

( ) 0di t R

i tdt L

+ =

( ) ( ) 0R

i t i t L

+ = (1.100)


38/44

40

( ) Rp tL

=

( )R

dt Rt LLt e e = =

(1.100) ( )t

( ) ( ) 0Rt L Rt LR

e i t e i t L

+ =

( ) 0Rt Li t e =

( )( ) Rt Li t Ce= (1.101)

0

( 0)i t I= = ; [ ( 0) ]0 0

R LI Ce C I= =

0C I= (1.101)( )

0( ) Rt Li t I e

=

t ( ) 0i t ##

1.19 RC

400 cos 2 .t V 100 210 .F

1.3

1.3 RC

( )q t (1.77)

( ) 1( )

dq t E q t

dt RC R+ =

1( ) ( ) Eq t q t RC R + =


39/44

41

( ) 1 ( ) 4 cos 2q t q t t + = (1.102)

( ) 1p t = 1( ) dt tt e e = = (1.102) ( )t

( ) ( ) 4 cos 2t t te q t e q t e t + =

( ) 4 cos 2t tq t e e t =

2 8 4

( ) sin 2 cos 25 5

t t t

q t e e t e t k = + +

8 4( ) sin 2 cos 2

3 5

tq t t t k e

= + + (1.103)

0 ; ( ) 0t q t= = (1.103)(0 )8 40 sin 2(0) cos 2(0)

3 5k e= + +

4

5k=

8 4 4( ) sin 2 cos 23 5 5

t

q t t t e

= +

( )( ) dq ti tdt

=

4 16 8( ) cos 2 sin 2

5 5 5

ti t e t t = +

(1.80) 16 8cos 2 sin 25 5

t t

(Steady-state current) 45

te

(Transient current) t

(first order circuit)


40/44

42

1.19

MATLAB

,a b 1. 43 2y y y x + + = 2. ( cos 2 ) 0y a b x y + + =

3. 6 11 6 xy y y y e + + + = 4. 2 0ivy xy y+ + =

5. ( ) 0d xy dx xy + = 6. ( ) ( )-x y dy x y dx+ =

7. 2 2 2 2 2a u x u t = 8. ( )2 2 2 2 2 2 2x u x x u t =

9. ( )2 2 2 2 2 2 2x u x x u t = 10. 2 2 2 2 2 2 ( , , )u x u y u z x y z + + =

11. 2 2u t xu d x dt + = 12. 2u x u u y v v x y + =

113. 2y xy= 14. ( )sin 2 (cos )dy y x dx=

15.2 2

3 (1 )y x y= 16. 3xdy ydx=

17. 2( )y x y y+ = 18. 2( 3 2)xdy y y dx= +

19. 2 2( ) ( )y x y dy xy x dx+ = 20. ( )ydx xdy x dy ydx =

21. 2( )yy xy x= + 22. 2-dx ydy x ydy=

23. x yye dy dx+ = 24.2x yxe dx ydy+ =

25. 2 2( 1) ( 1)y y x= + + 26. ( ) ( )2 22 2 4 3y y y x x= + + +

27.2

sin 0y y x + = 28. ( )cot 0xdx x dy+ =


41/44

43

29. ( )2 3 0xy xy dx dy+ = 30. ( )22 sin 3 1 cos 6 2 0x x dx dy+ + =

31.3

2 cos 0y y x = 32. ( ) ( )2

2 cos sec sin 0x y dx x y y dy+ =

33. 2( ) 1 0y y + + = ( y dy dx = )34. y y = 35. 2( )yy y =

136. 2 (1 ) 0 ; (2) 1xydx y dy y+ + = = 37. 2 sin 0 ; (0) 0y dx y x y + = =

38. 2 0 ; 4, 1xy y x y + = = = 39. 2 0 ; 0, 100y y x y + = = =

40. 2 ( 2 ) ; ( 3) 1xdx dy x xdy ydx y = =

41. (2 ) ; 1, 4dy x ydx xdy x y= = = 42. (2 ) ; 4, 1dx y xdy ydx x y= = =

43. 2( 1) (2 sin ) 0y dx xy y dy + =

44. ( )3 2 22 ( ) 0xy x dx x y dy+ + + =

45. 2 2(3 6 ) (3 2 ) 0x xy dx x y dy + =

46. ( ) ( )2 2 2 2 0x x y y dx y x y x dy+ + + + + = 47. 4 2 3(2 sin ) (4 cos ) 0xy y dx x y x y dy+ + + =

48. 2 2(2 3 2 ) 0y dx y xy dy + =

49. ( ) 2(2 cosh ) cosh sinh 0x xy dx xy xy xy y dy + + =

50.

( ) ( )

22 0y yxy e dx x xe dy+ + + =

51. (1 ln ) (1 ) 0; 0, 0xy dx x y dy x y+ + + = > >

52. (sin sin ) ( cos sin ) 0y y xy dx x y x xy dy + =

53. 2(2 1) 0 ; (1) 2xy dx x dy y + = =

54.2

(2 ) ( ) 0 ; (1) ln 2y y

xy e dx x xe dy y+ + + = =


42/44

44

55. ( ) ( 2 ) 0 ; (2) 3x y dx x y dy y+ + + = =

56.2 2

( ) (2 1) 0; (2) 2x y y xy y+ + + = =

57. (2 1) 0y xy dx xdy+ + =

58. 2 2( ) 0x y x dx xydy+ + + =

59. 22 (2 ) 0tds s s t dt + + =

60. 2 2( 1) ( 1) 0y y dx x y dy+ =

61. 2 2 2 2( 1) ( 1) 0y x y dx x x y dy + + =

62. 3 2 3 2( ) ( ) 0x xy y dx y x y x dy+ + + + =

163. 2(2 )y x dx xdy+ = 64. 22 exp( )y xy x x + + =

65. tan secy y x x + = 66. cot sin 2y y x x x + + =

67. 2 (2 1) 0x dy xy x dx+ + + = 68. 2(1 ) 2x y xy x + =

69. 2(1 )y y x x x x + = 70. 32 ( 1) ( 1)y y x x= + + +

71. (1 ) exp( )xy x y x + + = 72. 22(1 ) 1y x y + =

73.( ) 0x y dx xdy + = 74. 3 2xy x y=

1

75. ; (0) 2x

y y e y + = = 76. ; (0) 3x

y y e y

+ = = 77. 2 3( 1) ( 2 ) ; (1) 1x dy x xy x dx y+ = + =

78. 2

(1 2 ) ; (0) 3xy x y e y + + = = 79. 2(1 ) (1 ) ; (1) 0x dy xy dx y+ = + =

80. 2(1 2 ) ; (1) 2y xy x y= =

81. 1.2 60-Hz (Amplitude)110 V. 15 . 3 .


43/44

45

(1)

t(2) ( )sin 120A t A

(phase angle)(3) (Amplitude)(frequency)

82. ( )q t RC 1 1

( ) ( ) ( )q t q t E t RC R

+ =

( )E t E= 0(0)q q= 83.RL 2 , 25 HR L= = ( ) tE t Ae= A (0) 0i = ( )i t


44/44

46

1. Wylie, C.R. ,Advanced Engineering Mathematics, McGraw-Hill, NewYork,

5thed., 1982

2. Kreyszig, E. ,Advanced Engineering Mathematics, John Wiley & Sons, New

York, 6thed., 1988

3. ONeil, P. V.,Advanced Engineering Mathematics, Thomson Learning, Inc.,

5thed., 2003

4. , , , 1 , 25365. , ,

1.Chapter1 ODE.pdf

Documents

Transcript of 1.Chapter1 ODE.pdf