Recall Wave equation

UH C Uxx wax cueUlxio fixutixio gix

D Alembert's Salin to wave

eqnulx.tlII fixth t fix Ct

Xtlt1 Ee f g's ds

X Ct

h f

fix Io

u i x E l Effi x t e t t fix ch

t o

tea

t to

a

travelling in oppositedirections

Domain of dependence

UH t solves 4th 9UxWith UK is flX Uf x e glx

Find the largest interval on which

modifying fix and 91 1 Can changethe value of U 12.63

Uh b t flu 1 film

1 tf f glsias

SO Ulrio depends only on values of

fix GH in 1 16 Lo

i i

Higher dimensional wave eqn

Vibrating membrane

I't's

wave egni UH C ou

UH.y.tl gp 0 Bo

solve this using separation of variablesUK yet fix.y Git

oft aT GHI

04 if4 4 0

Boundary value

problem

solve this faka

New topic Sturm Liouville problem

Y t Xp o

461 414 0

di TET 4 _sis

Orthogonality h4mdx 0 Mtn

Ex Heat eqn

Cf Ut Koll x x 1 LUseparation of variables

41kt YING Ifs

p 41 1641 1 140 1 Gits1 241 3 Gif

lift th't'tw pdependson t depends on x

ko 4 t dyed 4 0

of lol 414 0

Boundary value problem

Y

Ex Radial heat flow

UK.y.tl UH O t

not depending on 0Ucr t

Ut OU Fmr r

Ucr tl o

unit o

uii a

separation of variablesUlriti ftp.G It

4 G'HF 444 Git

crop

Ty x

toll t dry o

Ofer Yury o

The general form of a Hurm Liouville

eqn is

P of't t 910 tar 4 0 a Eb

PHI 91 1 Text are fats of x

Pk 70 THI 70

Bcs holla 11,944 0

224lb Tfa 044 0

Goal Find eigenvalues daFind eigenfets 4h

othegonality of eigenfots

7hm all X are realI eigenvalues

X Cdr C 7g C

Each eigenspan is simplei e t din c eigenspace

g

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fix 7 E an fulxhit

othyonulity.com Ion Jab 9h9m dx o

if m th

An fab f H 9mxirix IxJab n'D rixidx