Lecture 15

( Undetermined Coefficients )

UndeterminedGoffs .

Gen.

Solution to Ky ] : a y'

't by'

tags O (homogeneous )

yhltl s C, yftltcayzlti

What about Ugg s SHI ? ( non - homogeneous ) IN i"

forcing

Idea : Suppose ypkl is a particular solutionfunction

"

( say found by guessing)Suppose yltl is another solution

So hyp s Htt and hcg ] = fat

Then L [ y - gp3 = Htt - tht so thy Linearity ! )

' So yltl - ypltl is a Solution to the homogeneous ODE.

Therefore, ylt) -

ypltl = C, y ,lHtGyjH for some Gts .

Thus,

LHS ypltl t c, y ,

Hit gym

We already know how to find y,kit gala .

{We only need to find Ypltl !

Special Case :

5- It ) s Re (pH ) et- ''htt ) for pH a polumomial

Examples :

Htt s 2ttt3t' (f so

,w:o)

Htt s ett

t It ) s Htt 1) e- It

5- It ) = cosftt - 3 sin HI = the ( HtHeit )this sin Cbt) s the ( - iebit)

Htt = at cos CHIT 7 sin Gtl tacos Gt ) = the ( HttI 2. ti ) ) edit )

If this the

ftp.tptt.i.tpntnlert

)1 2 fl

Try ypltl = Re

Xc,

t ft t . . . t cm.tn/ert}as

long as Vert ] 40

Is Vert ] so tryt ( C

,t ft t . . . t Cmt) ert C r

single root )

tfc,t ft t . . . t Cmt) ert Cr double root )

Example is. y"

t 2g 't 7y s 3 ett Try ypsC

,ett

= #7) e't

Try yps (Gtgtle 't

y'

't 3g 't 2g s eat

( roots.. . ,

,. , )

-

Trt Yp

' Hc,est )

y"

+3g'

tags Lte-2T Try ypstcc, tgt ) e- Lt

y "t4y'tyg s test Try ypstdlftgtle.ttCrs . 2 is double root !)

I 't 2g 't y = 2 cool 4th s the { 2e4itl.Try yps Ref (Citi g) e4it}

y'

't 16g s 2 sin Htt = Ref .2ie4it).

Try yp : Re { tcgtiglehlit}- t

y' 't 16g : e cost # s the { et'"iH I

Tng yp = the { C Gti g) elitist}

Example

payFind the general solution to

y' 't 2g 't y s

featSIN.

Char. poly . r 't anti has double root rs . I

.

Try ypltis Ae't IAest 't't HAITI 't Aest

Them Yj t Lyft gps 424+2121+1) A eats 6 dnt

or 9A s 6 ⇒ As 6452g 2pm 2g ett

Gen Sohni yttls3.edt t CC, tgt ) e

. t

3

(b) Find the general solution to y"

thy 'ty

= Kostal -3 sin lat

Sod Note : 2 cos Htt - 32in Cat ) s Re { Its;) edit)

Try ypltls the { A edit) .Set 2ps A edit

2- It Ttp't -2ps{ Gift 212Mt 's } A edit sfg+4;) A edit

Want Zp" t 22ft 2ps Gt3i ) edit.

Hence ( -374 ;) As Gt3i) ⇒

As sfg-

Ifi.rs?e-itaiHxdSoypHsRefl?g?ii)editf=fgcosktltLtgh

'm ht's

us lat . tan"Md )

General sohn : yhsktgtle.tt Ref it } .

Example .

Find the general solution to

y' 't 2g 't y s teat

SIN.

Char. poly . r 't anti has double root rs . I

.

Try ypltl = (

AtBt) ett

yp'

HI s

Beatt 2C

AtBtlett

= 42

AtBHt

2Bt ) ett

Lp"

It ) : 2

Bestt

2(l2AtBlt2Bt

) edt

= 44

At413) t

413f) eat

So Jp't 2yp'

typ s

(4At413t 4Bt) edt+ 242AtB ) t2Bt ) eat

+ ( A t B t ) eat= (9At613t 9 B t ) eat = tent

sogo.it

' ?;o ⇒InightB B ⇒ Lpkss ttq) eat

Gen Sohn is

yltls (c, tgt ) et t ( 4g

-

2g ,)e+2t

Example .

Find the general solution to

4ps y' 't 2g 't y s ( 4th ) et

SIN.

Char. poly . r 't 2 rtl has double root r , . I

.

So the general solution ni of the town

yltts K,t Stl e

'tt yplti .Tofind ypltl try

ypttl = t

'tATB f) Et

After a bit of algebra, we get

Llgpts (

2At6BE) Et s I 4th ) e

't

So 2 As 4 and.

613=8 ⇒ As 2, B s 41g

Thegeneral

solution is , therefore,

y His (C,tgtle.tt t42t 4ft) et

= { Gt St t 2kt 4g it .