(1) The order of ODE: the order of the highest derivative e.g., Chapter 14 First-order ordinary...

(1) The order of ODE: the order of the highest derivative

e.g.,

Chapter 14 First-order ordinary differential equation

order) (second order),(first 2

2

dx

yd

dx

dy

(2) The degree of ODE: After the equation has been rationalized, the power of

the highest-order derivative.

e.g.,

ODE degree second andorder third the

)( and )()(0)( 23

3

3

332/322/3

3

3

dx

yd

dx

yd

dx

dy

dx

dyyx

dx

dyx

dx

yd

(3) The general solution of ODE contains constants of integration, that may

be determined by the boundary condition.

(4) Particular solution: The general solution contains the constants which are

found by the boundary condition.

(5) Singular solution: Solutions contain no arbitrary constants and cannot be

found from the general solution.

with n parameters satisfies an nth-order ODE in general. The boundary conditions on the solutions determine the parameters.

xaxay cossin 21


14.1 General form of solution

),.....,,,,( 321 naaaaxfy

Ex: Consider the group of functions

equationorder -second 0

cossin

sincos

2

2

212

2

21

ydx

yd

xaxadx

yd

xaxadx

dy


14.2 First-degree first-order equation

0),(),(or ),( dyyxBdxyxAyxFdx

dy

Separable-variable equation

dxxfyg

dyygxf

dx

dy)(

)()()(

1)2

exp(

)2

exp()2

exp(1

2)1ln(

1

)1( :Ex

2

22

2

xAy

xAC

xy

Cx

yxdxy

dy

yxxyxdx

dy


Exact equations

dy

ydFdxyxA

yyxB

y

yxU

yFyFdxyxAyxU

cyxU

yxdUx

yxB

y

yxA

y

yxU

xx

yxU

y

y

yxUyxB

x

yxUyxA

dyy

yxUdx

x

yxUyxdUdyyxBdxyxA

yxU

)(]),([),(

),(

by determined be can )( and )(),(),(

ODE of solution the is ),( so,

0),(),(),(

)),(

()),(

(

),(),( and

),(),(

),(),(),(),(),(

satisfies which ),( function aFor


solution the is 2

3

2

3

)(0

)(2

3)()3(),(

exact is equation the1 ,1

),( ,3),(0)3(

03 :Ex

2

12

2

2

1

2

1

cxyx

ccyxx

cyFdy

dFx

dy

dFx

cyFyxx

cyFdxyxyxU

x

B

y

A

xyxByxyxAxdydxyx

yxdx

dyx


Inexact equations: integrating factors

)(1

)( })(exp{)()( if (b)

)(1

)( })(exp{)( )()(1

)(

)( if (a)

it find to solved be can )(or )( if (2)

it findingfor method general no ),( if (1)

)()(),(factor gintegratin an by gmultiplyin by

exact made be always can aldifferenti The

),(),(but 0),(),(

y

A

x

B

Aygdyygyy

x

B

y

A

Bxfdxxfxdxxfdx

x

B

y

A

B

d

dx

dB

x

B

y

A

dx

dB

x

B

y

A μ

x

yx

yx

Bx

Ay

yx

BdyAdx

x

yxB

y

yxAdyyxBdxyxA


cyxx

cyFyFyxyFyxy

U

cyFyxxyFdxyxxyxU

ydyxdxyxx

xxdxx

xx

yyxyx

B

y

A

Bxf

yx

By

y

A

xyyxByxyxAxydydxyx

x

y

ydx

dy

234

2'3'3

123422

322

2

22

is solution The

)(0)(2)(2

)()()34(),(

ODEexact an is 02)34(

)ln2exp(}1

2exp{)(2

)26(2

1)(

1)(

exact.not is ODE The 26

2),( ,34),(02)34(

2

32 :Ex


Linear equations

solution the is )()()(

1)()()(

)()(])([)(

)()()()((1) Eq.

})(exp{)()()()(

ODEexact an is 0))()()(()(

))()()(()(

)1()()()()()(

)(factor gintegratin multiply )()(

dxxQxx

ydxxQxyx

xQxyxdx

d

dx

xdy

dx

dyxyxPx

dx

dyx

dxxPxxPxdx

xd

dxxQyxPxdyx

xPxQxdx

dyx

xQxyxPxdx

dyx

xxQyxPdx

dy


)exp(2

)exp())(exp(2)2ln(

22

22

)2(2

method separated-Variable (2)

)exp(2

)exp(2)exp(4)exp(

)exp(}2exp{)( (1)

42 :Ex

2

222

2

222

2

xky

xkcxycxy

xdxy

dyxdx

y

dyyx

dx

dy

xcy

cxdxxxxy

xxdxx

xxydx

dy


Homogeneous equations

x

dx

vvF

dv

vvFdx

dvxvF

dx

dvxv

dx

dy

vxy

x,yfyxf

yxf

yxBxyyxA

yxByxA

x

yF

yxB

yxA

dx

dy

n

)(

)()(

onsubstituti the Making

)(),( obeys

it , anyfor If n. degree shomogeneou of function a is ),( (2)

degree. third the with and e.g., degree,

same the of functions shomogeneou ),( and ),( Where(1)

)(),(

),(

3322


)(sin)sin()ln()ln(sin

ln)ln(sinsin

cos

lncot

)tan(set

)tan( :Ex

1

12

1

AxxyAxx

yAx

x

y

cxcvdvv

v

cxx

dxvdv

vvdx

dvxv

dx

dyv

x

y

x

y

x

y

dx

dy


Isobaric equations

mvxydxxm

dyy

yxB

yxA

dx

dy

onsubstituti a make then , and to relative weight

a given each are and if consistent llydimensiona is equation The

),(

),(

cxxycxyxcxv

x

dxvdv

vxdx

dv

x

v

dx

dvvx

xx

v

vxdx

dvxvx

dx

dy

vdxxdvxdyvxym

yxdydxx

y

xy

yxdx

dy

ln2

1ln)(

2

1ln

2

1

122

)2

(2

1RHS

2

1LHS

2/2/1

lyrespective 1,2m 0, 1,2m is litydimensiona the 02)2

(

)2

(2

1 :Ex

222/12

21

2

2/12/12/3

2/32/12/1

2

2


Bernoulli’s equation

ODElinear )()1()()1(

)()1()()1(

)()()1

(

1)1(

linear nonlinear onsubstituti a make

1or 1 where )()(

11

1

xQnvxPndx

dvv

yxQnyxPn

dx

dv

v

yxQyxP

dx

dv

n

y

dx

dv

n

y

dx

dy

dx

dyyn

dx

dv

yv

nnyxQyxPdx

dy

nn

n

nn

n

n

343

33333

13

3

33

33

434

44341

43

6 is solution the

6)6(1

)1

()()()(

1)(

1}

3exp{})(exp{)(..

6)( ,3

)( with ODElinear 63

23

12

3

33let

2 :Ex

cxxy

ycxxdxxxx

dxxQxx

xv

xdx

xdxxPxFI

xxQx

xPxvxdx

dv

xx

y

dx

dvyx

x

y

dx

dvy

dx

dvy

dx

dy

dx

dyy

dx

dvyyv

yxx

y

dx

dy


Miscellaneous equations


)(

onsubstituti a make

)( (1)

xbFadx

dyba

dx

dv

cbyaxv

cbyaxFdx

dy

11

11

2

2

2

)1(tan

tan1

111

)1( :Ex

cxyx

cxvdxv

dv

vdx

dy

dx

dvyxv

yxdx

dy

ODE shomogeneou a

0)((

0)()(

shomogeneou is RHS and let (2)

fYeX

bYaX

dX

dY

gfefYeXgYfX e

cbabYaXcYbXa

YyXxgfyex

cbyax

dx

dy


2223

123

1

2

2

)32)(34()21

1)(1

1

14()1(

)3exp()2)(14()2ln(3

2)14ln(

3

1ln

23

2

143

4

472

42

42

472

42

52

42

52let

42

52

10642 and 0352

,let 642

352 :Ex

cxyxycx

y

x

yx

cvvXcvvX

dX

dX

v

dv

v

dvdv

vv

v

v

vv

dX

dvX

v

v

vXX

vXX

dX

dvXv

dx

dvXv

dX

dYvXY

YX

YX

dX

dY

YyXxyx

yx

dx

dy

14.3 Higher-degree first-order equation


0),().....,(),(),( is solution general The

,...2,1for ),( equation of solution the is 0),(

),( and ),(

0))........()((

for 0),(),(...........),(

321

21

011

1

yxGyxGyxGyxG

niyxFdx

dypyxG

yxFpyxFF

FpFpFpdx

dypyxapyxapyxap

n

ii

iii

n

nn

n

0)]1()][1([ is solution general The

0)1()1ln(ln1

202)1( (2)

0)1(1lnln1

0)1( (1)

0]2)1][()1[(

for 02)123()1( :Ex

221

222

22

2

11

2

22223

xkyxky

xkycxyx

xdx

y

dyxy

dx

dyx

xkyc)(xyx

dx

y

dyy

dx

dyx

xypxypx

dx

dypxyypxxpxxx

dy

dp

p

F

y

F

pdy

dx

dx

dyppyFx

x

1

for ),(

for soluable Equation


xyyxy

pxypyxpypy

ypyp

kkxyykxkyy

kx

y

ky

y

kpkpyc

ypcyp

dyyp

dpp

dy

dpy

dy

dpyp

dy

dpypypyp

dy

dpy

dy

dp

p

y

ppdy

dxpy

p

yx

dxdypyxppy

p

solutionsingular 038)6/1(94

9)16()3()6( equation. origional the Change

6/1061(2)

solution general 6303603)(6

1lnlnln2ln

2202 (1)

0)2)(61(12613

363

/for 036 :Ex

2322

22222222

22

23322

22

2

22

2

222

2

22


dx

dp

p

F

x

Fp

dx

dypxFy

y

),(

for soluable Equation

solutionsingular 0021 (2)

solution general 4)(

4)2()(eq. origion the intoput

1lnln

202 (1)

0)2)(1(

0)1(2)1(0)1(2

2222

02 :Ex

2

222

2

2

22

2

yxyxxp

kxky

kxxpykkxp

cx

px

dx

p

dp

dx

dpxp

dx

dpxpp

pdx

dpxpppp

dx

dpxp

dx

dpxp

dx

dpxppp

dx

dyxpxpy

yxpxp


Clairaut’s equation

ODE origional the in eliminate 0),(0 (2)

solution general )()(

)(eq. origional the intoput

0 (1)

0)(

1112

12211

212

2

ppxGxdp

dF

cFxcycFc

cFxccxccdx

dyp

cxcydx

yd

dx

dp

xdp

dF

dx

dp

dx

dp

dp

dF

dx

dpxpp

dx

dy

)( pFpxy

solutionsingular 04442

2

02 (2)

solution general ))(()( (1)

0)2(2

:Ex

2222

22

2

yxx

yxx

yx

ppx

ccxcFxyppF

pxdx

dpp

dx

dpp

dx

dpxp

dx

dy

ppxy

(1) The order of ODE: the order of the highest derivative e.g., Chapter 14 First-order ordinary...

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Transcript of (1) The order of ODE: the order of the highest derivative e.g., Chapter 14 First-order ordinary...