LecturerdlisiuosoiolalforaeigReaeUoowethugaboueteowplexhuube#

=

,?I .= 3tYI-6it8 =

'II - ¥ -Ei

polar forces of a complex hueber

Itai p= ¥1 Fague

I = Fsaoso 2=r5 Sine though

Itai = 55 oooo t i Fssiuo = is ( cos@ tisane) = Fse"O

F- = norm / absolute value leeeogceitude of Ita'

D= argument/ polar angle of Itai

Exaeipe y"

t 2g't 2g = 10 cos Cat)

homogeneous part : X't 2X +2=0 Atl )-

ti -o X = -i te'

f-Hilt = e- teit = e-+

( cost ti suit)

y , Ltte Et cost Yak) -- E- suit

we have deue somethingparticular solution isweeter when solving

Method I guess yp (e) = A aescze) t Bsuiczt)rn you- hang ,

blear eq

of order L.

Method 2 consider thee"

complexities deff eq"

Mr

locos Cat ) ~> toeit

.

Note that Re( leet") = he cosset

sit

y"

t 2g't 2g = eoe

2. it

guess y A)= Ce (c e-it)"

+2 ( cedi )'

+ zoe

"

,@ eat

→(edit + aice-ittzceiteloe.it -acidic t2C= to C-2 tai) Ce lo

C -1¥.

- if. -

=-5hL = - i - aiIta

So yeah C- l- 25) e"

f-I- ze) ( cost tesuize) =

= - cosset - e- Juice - a'

cosset t2dui2k= @said - coset) -i (sake tzcoszt) .

Susie our oagueel equation is the real part of the complexities eg ,

Yp CE) = Re Yale) = 2 Seize - aesze

So the general solution is

LCE)-- ypltltkiy , Celt kzyzltl-2suizt-coszttkietcos.tt Kz e-taint

reverts if the equation was y"

toy'

tag = rosiest ,then we wowed have taken thee cee~p.at of Yc cel,namely

,

- Sci2t - 2 cont as ypct).

tlowdocuegraphtleeadeetr.co#ypCeI--2Sceizt-cos2teeeigLebe diff cult to draw.

Do thee following eeeoteoedi back to Yale)

ya # = C- I - zit e-it=Fseio Iit n 0×180+63=2430

0 as in the graph #= jzeCOt2t) c

'

. - z

=

= Fs ( cos Cortot) t i sceicotzel )

so yp Ct) = is eosc2.tt O) .

this is equal to 2.suis t - adze .

⇒.÷. a-ease..

we want to draw the graph ypCE7= OF coscztto)-=rEaos(a G- + E) )So we deft flee graph of Fscosztto the left by E- a 24£ I 121%Inn

f- -

÷:#other ooeeetooas will www.rpetoypltt-TEcoscstto) as t -so

p? Note that the system is not autonomous

so ooceetrae curves will cietevaece.

put Val - y' le) as thee sad variable

old = V

⑦¥1! y " = -zu- y + cocosze

not autonomous.

particular solution is ypltl = OF eoscrdkto) Vplt) -ftp.lele-2fssceiczttolyp-rscoscztto) ¥p= - B- Sein Cueto)

⇒ ( yp )'t ( 12=5 it's an ellipse !

✓ A 205How do we feed thee direction?

when Itt0=0 ⇒ y=T5 V=0-55¥>yatte - E ⇒ g- o re -ers

L⇒ clockwise

•-255

Auotlee.ro#pe- y"

toy'

tag = suit

holeof solutions , I t 31+2=0 Xie - I 12=-2

cite cost t is wit So the complexities eg is

→since Reeitesint

y"

tag'

toy - eit and we'll take the ing part

guess a part enter solution i

y AseCe"

(ceit )"

t 3 (ceil )' tacet

e e"

- cel't t ziceit tzceit = eit - C tzic +2C = I

c- C ' tail - I C = ÷ . = Iif - to - Ioi

so ya lek ( of -fi) eit - ( of - i) ( cost t isait) =

= of cost t suit - fo i cost + sort =

= To cost t suit t i ( fo suit - cost)So ypce) = ¥ suit - cost

,and the general solution is

yet) = ki Et t ka E"

t fo (suit - scost)