Pages 10 11 of this lecture I

Lec will be done ou lecture Il

Differential equations partial derivatives

I Waveequation

ucx.tl Osx L t O

o L Eposition at pointf at the moment it

Three components

Equation off _IIIaInitial u x 0 f x osx chconditions Initial position

atCx o gCx Osx Ltinitial velocity

I Boundary ufo.tl sett t 0conditions all t they t 20

ft Picture

µqBdgrInitialconditions

x

I Heat equation ucx.tl osxs L

t O

Temperature at toilet Xo L

at the moment t

Equation c2 o exc Lf z O

r rr r n r

Initial conditionsuh 0 Glx o exc LInitial temperature

Boundary conditionago f qk uLot YtTemperature as theend hotels

Moreequations

9 Ifa O Laplace

one o

Efcx.es Poisson

More to follow

Maia notionsmm

Homogeneous equationzero right hand side

Linearity superposition principle

Ugly t uzcx.tn solutions of a homogeneous earnucx.tl L uacx t dzuzcx.tlalso solution

Solving wave equ

II c o exc L t o

2 u 0,11 0 all t O t 0 Fixed ends

3 u x 0 ft Ut x O gCx O Lx L L

Main ideasmm

Do NOT tug to meet all 7 27 3

Obtain MANY solutions of thehomogeneous part line 4,27 then addthem in a way to week 3

SeparationofuaruabtedStep 1 Find man c solutions of 1

Ju Ije CE of the form

ulx.tl Xcx Tfc

i i.tn Cx7 dTkIX4x1Tfc XCxl

ithett f L too

It 4EH

k ksome constana

X Cx kXlxI o oaxal It Kitty o too1 6 0 X L O

EndofstepistepIBoundary conditionsmm

El 4 oak TC 4 0 too

X x k X x o o ex c L

Which one to start with

2 X x LX x o X 61 0 X b o

Im

Qinghai.isPossibilitiesp k o k

ya 4 0 3 ke O ka p

choou k ph px px

X Cx p'Xcx XG Ae Be

Ho At 13 0 Xcx Ale e

XCL A e 4 0 A 0

Conclusion

f p NO nontrivial solutionsT y

cho.ie 2 k o 7XCx7 0 sXCx Ax

BXCoO 7B 0 sXex7

AxXCLO A _0Conclusion

NO nontrivial solutionsb DT K

THE last hose Choice 3

k ph X G p'Xcx O X 67 0,1 14 0

Xcx A siwpx tgag.pexH o sB o

Xcx AsingxX 4 0 ASimple p n'T h 1,2 I

Elementary solutionsme

kn Chip Xu Au whEx

t EkwT 0

me

Pu Lu

Tna duSimcha ftp.wcoseh It

T il 4Unix t µ Tule

A sinwIx qucos PfSim 7

End of step 2

u Tve have a lot ofelementary sobekothem

Steps Meeting the initial conditions

Idea Combine the solution from elementary owes

ucx.tk In Sinn x KloschittpusinekIt

mix o f x f i

that II Sinh x f due Simchat puchICaschEt

utcxo

q

Ea EsfcxlsiuhExdxkcfEIT.sfgcxs.ME

File this part ofthe page

Resume Steph

step 2

step 3

Analizing the elementary solutionsmum

m Siu'ExNormal modes Siu'Ex

Other boundary conditionsmm

Free end r

In this case the boundary

If condition Leske's tU lo t O

similarly af other end

C Exam presentation

More complicated cases

Utt Tuxxt'll 04 4 IT t O

mm ulo.tl _O ult t D

ncx.io SCH Utero GtxSteel x iX T XT

ucx.tl X TH

i k

It is more convenient to write

i x itsk l D

K I LO

Heat equation f dIuox

1 Where it comes from

9 EIt

Ix hHgt at next

ucx.tt ot Vx tI xlxtbx.tl uf.x.tl Dt

Fte sxt E

2 Setting the problem Various boundariesconditions

FTII if p 04cL too

re xp f x o exs L

A Fixer b.eu Coat D cell f O

B Insulated ewes u lqH o 6 1417 0

4 Educated guess I

A 1 c'II oexal tso alois 0 all.tl o

B II c eOsx L t O 6 194 0 4 6,4 0

ouestiou thigjelx.tk

At B 1

t.tl dlxx a

5 The same three steps

i ulx.tl Xcx 1fti He ik

7 ik T Oi tooi i

Boundary conditionerk Xiao ocx L X 6 so x L O

r e

f EXE Exam 2013 problem

al ft 05 592

Eax atEdie

Find Cos Fourier series

b Find all solutions of the forcesuh t X G Tle for the earn

let _U U Osx t20

with the boundary conditions

Ux loot 0 4 471 3 0

a Find the solution of salislyiuf.CAand such that uh o f x Osx E