Lecture 20

Into.to

LaplaceTransforms

Laplace Trusts• piecewise continuous For some M, a , and K,

+It) :. subexponential growtht fitness:f:i." " "↳

subexponentialgrowthtxamples.

-

L { D= ?

L feat} s ?

L{ caslbtt}s?

L{ sinlbtl } ' ?

tapeworms• Heat :[eat-stat

Angie, :( piecewise antinous =fgizf"e'"'a'tat

• doesn't grantor^ sat =¥n. H

5-HI✓ for s > a.¥ so

g{eat3'If#Is meat fort > KThen

,the Laplacetransform

i:÷÷÷¥÷:¥÷i÷÷÷÷÷÷:i÷÷÷:*.in.tianya.slo.se-stat =¥m."A)t÷s= 's.is

sting, fttettdt bassos.

=aliggfgsh) - ='s = statist

÷: so :i:"÷÷

Linearity

④L{afHtbgHB=aL{fHYtbLfgk#

• Derivatives :

④ Ll5HB=sL¥Lfsikf-SLHHI-sfld.si

5.dvingInitialValuePnoblemsusingLaplacetransforna y"t by't Cy s Elt) * LEH=sLH*{ gloss yo y'lo) s yo' *

LlstalsiLHaA.stios.sic@Erxample.y"- 3g'

+2g so④ LlsH=sL¥{ gloss 2 y'lol : 't Lfstal-ILHHI-sfld.si

• Applicationto initial value problem :

•Derivatives:

ayxtbjtcg.tk#Lfsttis=sLffkt-flD{ 2g%dg;

prog fo%ae-stdt-l.gg '

HE"dtLet ftp.LHHH

Integrateby parts) . 161=212411=algin fittest]! -fffktf.se#ThmafsI4sl-syo-giItbfst4sl-ydtcI4l= - flat fototttlestdt = FCS)

= LISH} - flo)OtohforIts. Lls sina.ss.io, .jo, tests tudou.IT#tbI

It : the #time aofdeaos :So Htt Lifted)

21*1=21*13 .Combing : we can she

sL{FAB-shotthe IVP if

=s{sL{ftp.floy-ggg.q are

① we can amputee =L{fHf

② We can compute L"IICD) .

.