2 6 Continuedor

⼆点贷品⼀嚼 anyFact i the soli to

IVPLY.ltf l t Yp It Y It Yiti 0is given by

Y It fig t.si foldswhere g l t is the sok to the homo IUP

Lg 0 g 10 0 g 10 0 ⼀⼀ 9 灬 o 0 g 1Therefore to find a particular so I n to Ly fone can first find g and then Compute Y byY It ffglt.si fastasg is called the Greasy associated to L

Ey Find a particular ol n to it cost tis intLy y y sin It if It i cost_isnt

Method Ii Key Identity Evaluation eit.it 2 is it2it sit eit eitfltr.ieÈezip12 22 l n i n i

L y fi e it zi is not a root of plzd 0L left p 12 i e2 it 3

ezitLLJ.ieit ie2itLy Ee2itLlffei2itj ie2itYpltlffle2it it fs.int

Method 2 i Green's function

Lg 0 910 0 S 10 1

M2 22 l n i n i

晶 点it C e ittc.it E cost t Es intg'm is it t E cost

10 E 9 tsintlz.CI

we may pick any ti say t 0

Y It ff g t_s funds sinxsnyiflcoslx yl.cosixtyE sinlt sj.si n 2s ds专 lflcosllt si zsi cosllt.si 2⼋ dstil cos It 351 wslttsDdsifsinlt 3SIE tsinlttsiilsitztl sintj tlsinzt s.int

fsinz.tt fs.int7Non homo.eq.w

variableweffsLltIyfn alt t tGn i It㗠 taltly f lt i_________11

Assume we know a funda set of solis Y 只to the homo eq.LI t 0

Then the general soI n to 1 is

y It Ypltltc.Y.lt t tCnYnlt1

We need to find YpHI

For simplicity we consider n 2

L It y y t a.lt y t cut y f l t____ 12

Know Y Yz want to find a particular s.li to 2

Try a so I n of the form

y l t U.lt Y.lt tuzctlY.lt Variation of parameters

y uiY.tu.it Ui h th Yiinpose condition Ui Y t u i h 0

y u Y t mi

y u i i u Y t ui Yi tui

⼼ y u i h t u Y tuiitu.ita.lt u.Yitmilta.lt u Yu.YyRecallLltlY YitG.ltlYitaltiY ol.lt y u i Y u i Y fitY h o

Yi Yi Hǚff memorize thisdet Hi剡 ⼆ WrCY 灯 tous solve ui.nl then integrate toset ui.cn

⼆ Find general soI n to

Lit y y ⼀ 蕊 y 志 y 1given funda set of soI as to l.lt y 0Y t ⽕ E 1

y It u ⼼ Y.lt tutti htt

ii 剡咋 倒 iiiIt'll 剡州 昨在怾 in

one zxzhdet2t.lt ii 1 tyxo

庣

1linearin u i t 幽 志2

u.ES蕊 dt f l 1 2 l d t t 2 taitIdon'tneed tc

u 词法 d t t s t d t h l r l t h 1 E 1v 1⼗⽇

driztdtapcicularsoli.inIt ⼀ ttztaitlttfhllttl.lt 2 1General soI么 yltkl ttztaitlttflnllttl.lt 2 1

C.tt Glf 1

Ruiwen Shuln(1+t^2)

Ruiwen Shuln(1+t^2)

Ruiwen Shu

Ruiwen Shu