Lecture 07

Modeling Examples

Review.

:

• General Solution to y 't pH) y = SH )

guys e-t 't ' felt't't Hdt t C e-

" t '

- in

2pm yhttlwhere BHI = fpltldt .

• Special Case : y'tky-ebtbs.es

Try Lpltls Aebt : Ib th ) Aebtsebt

⇒ As big So ytttsbiqe-bttce.tt

Examples .

- Gen.

Soh to y 't 5g : ett ys ett t Ce 't

7- -15

-Gen

.Soh to y 't 5g =P y : I t Cest

- Gen Soh to y'

tty s e's 't

Try ypltts At Est ( Ate 't ) 't 5C Ate 't )= Heist = e

't⇒ Asi

Gen Soen : gas test t Ce't

Example 3 . y't 3 y = cos C wt )

Try yplt) = A cos Cwht t B sin I wt )

Thenyep

't 3 Yp =

fw A sin htt t WB us hot)) t 3 ( Aws C wt ) t B sin C wt ))= ( 3A t w B) cos I wt ) t I 3D - w A) Lin hot) = Cos hot )

So3A t w B s 's

{- w At313 so

⇒As Fw .

B ' Fw .

So ytt) : Ftw .

cool whtg:-p

tin hot ) t Ce' "

= Gwyn cos l wt - g) t C Est , tangs Wg

Example.

deaf thy = cos C wt )

Example .

dy.

atthy -

- coscwt )

LH=ypHtCe-StTry yptD=AaoCwHtBsinCwt )

Theny; # + gyp # =

w - AsincwtltBaskettthlfaaocwttbsinlwtt)= @Btkhcoscwtlt

C- wAtkB ) sincwtl-cos.cat)

⇒ AskB=wk4w' ' either

So yplttshk-wcoslwtltsw-wsinlwd-fywcoscwt.pl ,

where tomy

=D.

s

Consequently,2 #

=¥wqusCwt

- e) +

Coifsteady k > o transientstakeSan

.

k④

Vote It dyqth.gs a,

cos C wit )

tacos

C wat)2

--

thenya ) : C e- est

+a

,

Vkatwpcoslwt. g) +

a

-

Veritycohost.q )

4,

stan " ( w.lk ) -

where { gas tan l walls)

Example .Find a particular solution of

y 't hyssinlwtlSolh . Try yp = Acpcwtlt Bsinlwt )

Then yjtlsyps lol - Asincwtlttdaslwt) )Is ( B sinlwtlt Aascwtl )

= Sinnott

⇒ Yes::%:i⇒%:* G-b' the

'

So Lps coslwtl t b- sinlwt )

h 't w' hi '

tw'

←with

4=4t 't Recall

= ÷w us ( wt - Y)

⇐Told's'm " . "s)

&p÷w.sincwtyi9tangs

wq

Low pass filter

> Mr

* i.JiYH = RI t

¥.

an I : dy V sq,at

e a

% c. v• light s R day t g a

- ,

0 At a

•

YltsrdkvdtkAt

Rc du

£t K =

Ugh

du

£ tack = £4

DV

at' tack = if FH

o

k IRC Is Vs It ) = cos C wt ) then

Vishwa' n'

Firmavolts .

.either . '-_ coslwt - y)

Vl t ( Rcw )'

Alw): YVltCrcwT called"

gain"

Example . 12=15 Or

C = 10×10-6 I

@qd00Hzfi5opi5a4000HzVHscEtkct4i5dfqiws.c

oslw,

't - ylthis)jqy.us/yt-gl--ceithe +y2iI⇐,uscwit - y)ty ,if rays

must ' Y)

I 1.sea eoslw,

t - g) t ( 0.03 ) coolwat - y )

0

10-3 see

Q

I1. 00×10-2

-2 see

Changing Units I Examples) Ebt

• Newton 's Law of Cooling . daffsdgjdgI.sk DIdit .

- time in Seconds

dfttbtskth.lt' T ': temperature So kdagptktskl.lt)

Units eyes :I

( in degases C)See

.

Matt) i ambient or

Yes natural timescale .

temperature .

=

IT TT =

Tittles)"

time constant'

di A

hett=thus) (dimensionless)

test . ( Yes)" time in multiples of Yb "

• Rc - circuit MrR

d¥t Lets fetch VIANa

RC is"

time constant"

units of RC = seconds tqdqkttjkstnctttilet Ts type or ts in . Rc or

Jts

tire )

dir

djksdatdtf s feats

Ifthe -

. Terry

at DT At dd¥tT= Tattled