Differential Equations

Ordinary differential equation (ODE) , Partial differential equation (PDE)

0// ,04 2222 yuxuyy

Order : highest derivative in equation

Linear equation, nonlinear equation

0 ),()()( yyyxryxgyxfy

Differential equations

Ordinary differential equation (ODE) , Partial differential equation (PDE)

Order : highest derivative in equation Linear equation, nonlinear equation

0// ,04 2222 yuxuyy

0 ),()()( yyyxryxgyxfy

Differential equations

Homogeneous equation, nonhomogeneous equation

Implicit solution, explicit solution

)()()( ,0)()( xryxgyxfyyxgyxfy

)1( 0),( ),2( )( 222 yxyxGeyxgy x

Differential equations General solution, particular solution

Initial value problem, boundary value problem

Exact, approximate, and numerical solutions

xx eycey 22 2 ,

)()( ')('

)()( )(

00

00

bybxyyxy

ayaxyyxy

First order differential equation, I

Separable equations;

Equations reducible to separable form;

cdxxfdyygdxxfdyygxfyyg )()( )()( )()(

cxyxdxdyxyy 22 22/9 4y9 49

xcxyxcucxu

x

dx

u

udux)-uuu(u

x

y-y

x

y

//1 /1 ||ln)1ln(

1

2 012 012 0xy-y2xy

2222

22

222

x

dx

g(u)-u

duugxuuyxyuxygy )( / )/(

First order differential equation, I

Exact differential equations;

Integrating factor; if ODE is not exact, it can be made exact.

xNyMyuNxuMdyyxNdxyxM / / /, / ifexact is 0),(),(

4/444/

)(/0)4(04

xcycxxyucxkykyxu

xkxyuxyuxdydxyyyx

cxyxydydxxdyxydxxdy 0)/()0)(/1(0 2

First order differential equation, II

Linear first-order differential equation (homogeneous);

Linear first-order differential equation (nonhomogeneous);

0)( yxfy dxxf

ceycdxxfydxxfydy)(

)(||ln)(/

)()( xryxfy

crdxeeycrdxeyeredxyedrefyye

dxxfhexFdxxfFdxxdFfxFyrfyxF

dyxFdxrfyxFdydxrfyxryxfy

hhhhhhhh

xh

/][)(

)(,)()(||ln/)()(/)])(([

0)())((0)()()()(

xxxxxx ececdxeeexyxfdxheyy 222 )(

First order differential equation, II

Method of variation of parameters General solution of homogeneous eq. is given

by

Idea is that we may replace the integration constant c by u(x)

dxxf

cev)(

)()( xvxuy

cdxv

ruvrurvurfvvuvu /)(

Ordinary Linear differential equation, I

Second order linear differential equation,

Theorem (Superposition Principle): Any linear combination of solutions of the homogeneous linear differential equation is also a solution.

)()()( xryxgyxfy

0)()(

)()()(

22221111

221122112211

gfcgfc

ccgccfcc

Ordinary Linear differential equation, I

Homogeneous 2nd order ODE with constant cocefficients

Try Characteristic equation (or auxiliary

equation)

Roots: Solutions: Examples:

0 byyay

) ,( 2 xxx eyeyey

02 ba

)4(2

1 ),4(

2

1 22

21 baabaa

xx eyey 2111 ,

02 ,0 ,02 yyyyyyyy

Ordinary Linear differential equation, II

Two functions are linearly dependent if they are proportional.

Theorem: If y1 and y2 are linearly independent solutions of ODE, the general solution of ODE is

If 1 2, general solution is

If 1, 2 are complex conjugate, the solutions are complex

t)independen(linearly ,(ii) ,dependent)(linearly ,3 (i) 22

121 xyxyxyxy

)()()( 2211 xycxycxy

xiqpxiqp eyeyiqpiqp )(2

)(121 , ,

xx ececxy 2121)(

Ordinary Linear differential equation, II

Real solutions from these complex solutions by Euler formulas:

Corresponding general solution:

Example: initial value problem:

)sin(cos)( qxiqxeeee pxiqxpxxiqp

qxeyyi

qxeyy pxpx sin)(2

1 ,cos)(

2

12121

)sincos()( qxBqxAexy px

1)0( ,4(0) ,0102 yyyyy

Ordinary Linear differential equation, III

Double root case (critical case)

Second solution by method of variation of parameters

Corresponding general solution:

2/ ,042 aλba

x

ax

xeyxuyuyuayyuyayayu

xyxuyeyyayay

2111112

11

122/

12

00)2()4/(

)()( , ,04/

xexccy )( 21

Ordinary Linear differential equation, III

Example:

Summary: For the equation

1)0( ,3)0( ,044 yyyy

Case Roots General solution

I Distinct real 1, 2

II Complex conjugate

1=p+iq, 2=p-iq

III Real double root =-a/2

xexccy )( 21

0 byyay

)sincos()( qxBqxAexy px

xx ececxy 2121)(

Ordinary Linear differential equation, IV

Cauchy equation( or Euler equation)

Try y = xm

The general solution is

Example: Critical case:

02 byyaxyx

212 , :roots two0)1(0)1( mmbmambxamxxmm mmm

2121)( mm xcxcxy

05.15.12 yyxyx2)ln()( 21mxxccxy

Nonhomogeneous linear equations

Theorem: A general solution y(x) of the linear nonhomogeneous differential equation

is the sum of a general solution yh(x) of the corresponding homogeneous equation and an arbitrary particular solution yp(x)

)()()( xryxgyxfy

)()()( xyxyxy ph

Nonhomogeneous linear equations

Method of variation of parameters:

Example:

ry

ry

Wv

u

rv

u

yy

yyryvyu

yvyuyvyuxyyvyuyvyuxy

xyxvxyxuxyxycxycxy

pp

ph

1

2

21

2121

21212121

212211

10

)0 settingby ( )()(

)()()()()()()()(

an Wronski:21

21

yy

yyW

dxW

ryydx

W

ryyxydx

W

ryvdx

W

ryu p 1

22

112 -)( ,-

xeyyy x 2