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
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