중앙대학교 건설환경플랜트공학과 교수

김 진 홍

- 4주차 강의 내용 -

1.4 Exact ODEs, Integrating Factors

We assume has continuous partial derivatives.

- If then

⇐ not only necessary but also

sufficient condition to be an

exact differential eq.

or

),( yxu

cyxu ),(

),(0),(),( Ny

uM

x

udyyxNdxyxM

yx

u

x

N

xy

u

y

M

22

,

)()( N

y

uykMdxu

)()( M

x

uxlNdxu

x

N

y

M

0,0

dy

y

udx

x

ududu

Ex 1) Solve

exact differential eq.

O.K.

Ex 2) Solve

exact differential eq.

IC.

0))cos(23()cos( 2 dyyxyydxyx

)cos(23)cos( 2 yxyyNyxM

)sin(),sin( yxx

Nyx

y

M

)()sin()()cos()( ykyxykdxyxykMdxu

)cos(23)cos( 2 yxyyNdy

dkyx

y

u

cyyyxyxucyyk 23*23 )sin(),(

0)23)(cos()cos( 2

dyyyyxdxyxdy

y

udx

x

udu

2)1(,0coshsin)1sinh(cos yxdyydxxy

xyx

Nxy

y

Msinhsin,sinhsin

)(coshcos)(coshsin xlxyxlxdyyu

,2)1( y

*)(,1 cxxldx

dl

cxxyyxu coshcos),(

x

N

y

M

yydy

dk23 2

x

N

y

M

1sinhcossinhcos

xyM

dx

dlxy

x

u

358.0coshcos),( xxyyxuc 358.011cosh2cos

- Reduction to Exact Form, Integrating Factors

Ex 4) is an integrating factor of

Several integrating factors exist such as

2

1

xF 0 xdyydx

cxycx

y

x

yd

x

xdyydxFQdyFPdx

,0

2

)(

1,

1,

1222 yxxyy

* may not be exact. But, if can be exact

by multiplying , then is called an integrating factor.F ( , )F x y

0),(),( dyyxQdxyxP 0FQdyFPdx

- How to Find Integrating Factors

* Condition for exactness of is,

If depends only on , is an integrating factor.

, where

If depends only on , is an integrating factor.

, where

x ( )F F x

y * *( )F F y

R

*R

Let , then and

Thus,

0FQdyFPdx

xxyy FQQFFPPFx

FQ

y

FP

,

)()(

)(xFF 0yF dxdFFFx /'

,)/( xy FQQdxdFFP

Qdx

dF

Fx

Q

y

P

1

dxxRxF )(exp)(

dyyRyF )(exp)( **

)(1

,1

x

Q

y

P

QRR

dx

dF

F

)(1

x

Q

y

P

QR

)(1*

y

P

x

Q

PR

xQFQdxdFyPF /)/(/

)()()( ykxyeykdxyeu xx

Ex 5) Integrating Factor and IVP

; not exact

; impossible since R depends on both and

; OK.

∴ exact

(general solution)

IC ;

x y

Solve

(particular solution)

1)0(,0)1()( ydyxedxyee yyyx

1)(1

)(1*

yyyxy

yyxyeeee

yeey

P

x

Q

PR

1/)(,1/)(,0)()( xexyyedyexdxye yxyxyeyF )(*

yyyyyxyyx exexx

Qyeeeyee

yy

P

)1(,)(

)(1

1)(

1 yyyyx

yeyeee

xex

Q

y

P

QR

,yexNdy

dkx

y

u

cexyeyxu yx ),(

72.3 yx exye

ceuy 72.301)1,0(,1)0(

*, cekedy

dk yy

1,00)( QrpyPQdyPdxdydxrpy

1.5 Linear ODEs. Bernoulli Equation

; linear ODE

; linear

; homogeneous linear ODE

-

-

if ; trivial solution

- ; nonhomogeneous linear ODE

, where

⇒

⇐ integrating factor

If is multiplied to eq.

If the above eq. is divided by and letting

F

)()(' xryxpy

xxxyyxxyxy sectan'sincos'

0)(' yxpy

*)(ln,)( cdxxpydxxpy

dy

dxxp

cexy)(

)( 0)(,0 xyc

)()(' xryxpy

pdx

exFppx

Q

y

P

QR )(,)0(1)(

1

cdxreyereyepyyepdxpdxpdxpdxpdx

)(,)'()'(

pdx

e pdxh

pdxhcrdxeexy hh ),()(

)()(' xryxpy

Ex 1) Solve

Ex 2) Solve

xeyy 2'

1)0(,2sintan' yxxyy

xxyccy 22 cos2cos33,12111)0(

Ex 3) Solve 22 63 xyxy

xxxxxxxhh ececeecdxeeecrdxeexy 22 )()()()(

xpdxherp x ,,1 2

xxdxpdxhxxxrxp seclntan,cossin22sin,tan

xxccxdxxcrdxeexy hh 2cos2cos)sin2(cos)()(

xxxxrexexe hhh sin2)cossin2)((sec,cos,sec

3222 3,6,3 xdxxhxrxp

333

2)2( xxx cecee )6()()( 233

cdxxeecrdxeexy xxhh

Ex 4) Solve 2)1(,1'2 yxyyx

xexdxx

hx

rx

p h ,ln1

,1

,1

2

2

11'

xy

xy

x

cxcdx

xx

xcrdxeexy hh ln

)1

(1

)()(2

2)1( cyx

xxy

2ln)(

Ex 5) Solve 12' xyy

2

,2,1,2 2 xh eexxdxhrxp

)()(22

cdxeexy xx

222 xxx cedxee

- Reduction to Linear Form. Bernoulli Equation

(a; constant) ⇐ Bernoulli Equation

If a=0 or a=1, the above eq. is linear, vice versa.

Let

; linear ODE

Ex 4) Solve

; linear ODE

Since

⇒

ayxgyxpy )()('

))(1()()1()1(',)]([)( 11 aaaaa pygapygyyayyauxyxu

2' ByAyy

,' 2ByAyy

AuBAyBByAyyyyu 1222 )(''

BAuu '

ABceuyyu

Ax /

11,/1

gapuau )1()1('

))(1( puga

,, BrAp

AxAxAxhh ceA

BcBdxeecrdxeeu )()(

AxAdxh

11 yyu a

)~,(

1'~),('

yxfyyxfy

1.6 Orthogonal Trajectories.

- orthogonal trajectories ; a family of curves that intersect a given family of curves at right angles

- angle of intersection ; angle between the tangents of the curves at theintersection point

* orthogonal, perpendicular

* qui-potential line, stream line

- ODE of the orthogonal trajectories,

is

Ex 1) Find the orthogonal trajectories, cyx 22

2

1

x

yy

y

xyyyx

~2'~

2'0'2

x

dx

y

yd2~

~

2*~,ln2~ln xcycxy

*22

2

1~,~~2~2'~ cxyxdxydy

y

xy

Ex 2) Find the orthogonal trajectories,

Ex 3) Find the orthogonal trajectories,2kyx

2cxy

cxdxdy 2x

ycxy

22

xdxydyy

xy 2~~

~2

'~

kydydx 2x

y

kyy

22

1

Cy

x 2

~22Cx

y 2

2

2

~

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