This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Math 427k ordinary, partial diﬀ. eqns.: defs
linear d.e.’s
order of a d.e.
solution, general solution of a d.e.
direction ﬁeld
1st order linear:
y + a(x)y + b(x) = 0
µ = e a(x) dx
use of deﬁnite integrals for init. conds.
existence and uniqueness for:
1st order linear
general 1st order
separable d.e.
exact d.e.:
M + N dy/dx = 0
My − Nx = 0 for exact
φ = M dx + h(y )
φ = C is solution
integrating factor µ(x, y )
[My − Nx ]/N = function of x
implies µ = µ(x) = e [My −Nx ]/N dx
[My − Nx ]/M = function of y
implies µ = µ(y ) = e− [My −Nx ]/M dy
homogeneous: y = F (y/x)
set v = y/x, becomes separable
nd
2 order linear
existenceuniqueness
homogeneous (second meaning)
linear operator L
fundamental set of sols
Wronskian W (f, g ) = f g − gf
connection between Wronskian and existence of fundamental set of sols
linear independence, and connection
with Wronskian
reduction of order
(homogeneous) constant coeﬀs
particular sols of inhomogeneous d.e.
variation of parameters
review of sequences and series
power series
radius of convergence
ratio test for radius
diﬀerentiating and integrating series
Taylor series, uniqueness
ordinary point
sols near ordinary point
regular singular point
sols near reg sing point
boundary value problems
L(y ) = λy
ay (x1 ) + by (x1 ) = 0
cy (x2 ) + dy (x2 ) = 0
eigenvalues, eigenfunctions
partial diﬀerential equations
separation of variables
heat equation
Fourier series for interval [−a, a]
cosine and sine series for [0, a]
variation of above
wave equation
Laplace equation
numerical solution of o.d.e.
Euler method
RungeKutta method ...
View
Full
Document
This note was uploaded on 09/18/2011 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Fonken
 Integrals

Click to edit the document details