differential equations. ordinary differential equation (ode), partial differential equation (pde) ...
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Differential Equations
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Ordinary differential equation (ODE) , Partial differential equation (PDE)
0// ,04 2222 yuxuyy
Order : highest derivative in equation
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Linear equation, nonlinear equation
0 ),()()( yyyxryxgyxfy
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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
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Differential equations
Homogeneous equation, nonhomogeneous equation
Implicit solution, explicit solution
)()()( ,0)()( xryxgyxfyyxgyxfy
)1( 0),( ),2( )( 222 yxyxGeyxgy x
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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
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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 )( / )/(
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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
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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 )(
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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 /)(
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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
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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
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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)(
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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
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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
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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)(
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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
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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
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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