Post on 13-Jan-2016
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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