Differential equations and Fourier Analysis summary notes

If the point is ordinary i.e
Gg and RSI is analytic attire

slide 23
independent solutions
If X Ko is ordinary we can find two
at x o
f the form

y E ane o
y E nana

y E nonpanic



If of the form t do I potato Euler equation

y P
slide 25
This will have solution of the form
y Satisfies Quadratic

t Bati p'tCho 1 p Bo o
Pi P distinct
y Axe
Pip Repeated Axe Bat Ink
y
t.am t iiiii
r a
aemanan
EE tee ao
from excessive term



ÉÉ
Y
Wh eat often

hi
method yo fansite ao assumption



É
frobenius

x i singular
guapon Éo let qiao in method

singular be showing Ragin I unianginipmesealant
Then show its regular
2 series
Two value fo p a recurrence relation solutions
converse ti anti befigured
onevalue of p



IÉ ÉÉ
looktnÉbnx



i
Egansent e
y ye Foxy

t
angular at x Il Has ordinary point at sea
so series solution
or a non integer i a y 2xytacaeny
26
solved
by

L

, consider when a integer

General solution to 66 for a integer at
pinIl
goals are singular

non integer APalatBQ x
infinite sums radio convergence
J
singular at 21 11 Here Qacx is nonpolynomial
2 integer legendre polynomials and is singular at 20 41


Legendre polynomials with negative indices



Pnee Ph Gc for n 1,2 3




functions
or 2T periodic


ancoslnxltbnsinln.cl
fac Et
not
for all
If
a cocoho dx
an


bn ftp.tculsinlmcldx

odd function even function
co na Cign
fl k fay f x fax

M Wh M Iiw

, Tip the second order differential
determine what form the solution of
To
0
equation is in
play tack y truly at e x
or
see if it the ODE is ordinary singular
I
regular singular


E.anx ygganec.sc y

Radius of convergence test


ftp converse ftmo Y 71 Conveyed

us
comparing
the term

in the series solution
him IÉ I y am
y dank




Note If a function contains 14 then it means the function
taylors
non analytic at since we can't write flat as a


Series
flat flat float
x
feat let not differentiable at x o