Review Session0810312020MA 266

pre blew 1

Laplace transforme t <3

= > gets usG) t'

get = ↳ t > 3

Llgctil - LLusCt1) formula exists

2) Original definition

↳Htt's Usct) ( Lt -354 6 Ct - 3) + 9 )↳ ↳CHIA -354Gt -9 )f

= UsCtl ( (t- 3143Gt- 3 ))

UsLt) (Ct- 314 6 Ct - 31+18 - 97 = ugh (Ct- 3 ft 6 Ct-3 I t 9 )-

✓singer formula : L LUs Ct) t'

I

= LLUs CH Kt- 3 ft 6 Ct- 3 I +911

= LLu,CH Ct- 3121 + 6 LfUsCH Ct - 31 I

+9LL UsCH I

= e-"

↳ +6g +9g )

problem 2

£1

.si#nlLLf*g1=LLf1LLgyysincatpl--sinLasp-icesasinp£1 2

t

egg ,

} = Joetsinzct- E) IT

¥×¥y =)!eT(sin2t case -cos2tsinzqjd-L-sinztf.e-coszcd-b-cosztJ.tets.intDEb b - -et sinzt integration integration>f by by parts by pants

problem 3

x' spx p . G'

j'

) xcel =/ ! )

Eigenvalues : (X - 1) (X - 3) + I = e ⇒ 4=2,2

p - 2e . C' "

, ) = > vi. vase = > v= EYE"Ifr

(MHzent;

xzut-etcv.tt Vz)

In order to find Va : pl - 22172=0

note:* - vi. fi fi 7) =L: : )

=> angrywould work → lets use vz . (d)

In order to hind Vi : PL- 22 )ve Vi = > Vi=

'

! ) (j)

⇒ acts . (g) Et- (Y)

xzctl .µ) t, (f) left /The linearlyndepeadatSolutions

General solution is XCH = Ci X , CH t czxact) # I

weened to hind Ce and Cz :

(f) excel - af! ) -14:) 1rem * I

→ [9+4=0

= > a. cel

C, =I

particular seluh.eu : XCH --'II )s

Problem 4

**) x's. pxi-fctip.LT ,:) Httt) - tf :/ →+

x.→→er,

Eigenvalues : (1+210+2) - I = 74-47*3=0 -t

Xo - I → evz

Initial guess : xp Lt ) = t + (g)

we need to plug Xp CHand xp

'

Ct) into ** I

e. t.E.IE::HH⇒ Efi::. it ⇒ E÷÷%

⇒ xpat = 44 x 's ["

s )JUsingen method etundetermined ceelticients

problem 5

F'

Cs ) = 's2- I

what is htt = £ LHS l l ?

*,Remember : LL- tfut I = Ics ) ⇒ fat , = - t £LFIsil

so htt = - It I'

I I = - ¥ I'

l s+. , I **I

= - I £ Lust, -2¥, I = -¥ E'

' ¥ E'

I I

=-It et-zte-t.ee

2T

problem 6 .

x' =p x P =L; ? )Form a fundamental matrix Q Ct )

Use theeigenvaluemethod

X, =-Z

Eigenvalues : ( x -4) ( X ti ) - 6=0 (×, S

X-

- - 2 V, = ↳ ) => x. CH = f.

'

g) est

tres vie ( Y) = > xzctl .- filet

PCH - faut Eti ) = II]You can also fern ettwhich is also another fundamental

matrix

problem 7

x' spx p.- L! ! )Eigenvalues : III → hind eigenvector

⇐ CH , htt) ) ka

⇒ xiti-xieh-etcat.ci ) ④-x.As t →a , Ehmdivs increases

spiral source

IF p= f? !)Lpesinie real party

Eigenvalues : - I ± i"A

xicti-xichsettcci.ci ) ¥-x.

As tea, the radius convergesatespiral sink

(negatived part)

p= LI ? ) Eisemann:L, Lise.

A

x.IT . ' ¥-center point(real part is e )

problem 8 govt)

y"

t tf = sin variation d- parameters

charm. Eg : r't 4=0 III!!; = > get sint

Ye

www..de#:..II 1=2

Y -- U, CHL,LH t Hut) YEH

u,= - J Twiggy, dt = - J sinztxzk.net = - it , c ,

1-Cz

ur = ) qq.gg#,dt=Jces2tIIsintdt=tzfces..n2t-tdt=lqlnlsin2H

⇒yltl-uiyi-uzyj-fzt-4.ceszt-fqlnlsinzti.cz/sinztb = aces 2t-czsinzt-l-ztceszt-flnlsinztls.int-

general sentient particular

selection hemegene.us Solution

4-henhemegenay part

problem

problem 9

did =3t2cL. I )

LOL = - I

Jay -is dy = JL3t44tt2) dt

y'- 2g = t3+2t42tt C

(y - D'

- i = t't 2tzt+ C

New ycel = - I gives c =3-

⇒ Cy . it ! t' +21-2+2++4 = > yet) = I

- ✓ t't 21-44-+4particular seluhien

problem leExactness

Etty'

)dtx2tydy=o (xxx)--

f g

2¥22 37--22 ⇒ sim3yI=2I⇒e2E

E. Jetty')dt= titty! ay )workmen 2£y=2ty

So ¥y Lt't tytc.gs ) = 2ty⇒ 2ty + C' Cy )

=

Zty=> c'Cy) - e

= > ay )= CCendant

= > the ty C centering a general Selulienehfxexxy

problem 11

2g"

+ y'

+ 22=8 Ct- S ) y lol = y'

61=0

Usingaplace transform:

25701+5761+2751=6"

- 55I

⇒ Yes ) - e - => ycH=£lusll=4+45+79ggusctie.tt#sinLfLt-si)-UsCtIeE4sinLft)

problem 12Substitution method

Y 'sT! g. ty= > Titty . 'g v. y

' '

y'

Finer

Final solution : j# Jet'dt + CET-

general Selah'

en