MATH 231, Worksheet

Date: May 8, 2020

Find the inverse Laplace transform for each function.

1. F (s) =e�⇡s/2

s2 + 9

2. F (s) =1

s2(s2 + 4)

3. F (s) =s

s2 + 6s+ 11

4. F (s) = e�s s

s2 + 6s+ 11

5. F (s) =1

(s� 1)3+

1

s2 + 2s� 8

6. F (s) = e�2s 1

(s� 1)3+ e�s 1

s2 + 2s� 8

Overriding Theme in course made possible by a focus on linear problemsTake a homogeneous version of the problem at hand

solve itsoIn often contains freedoms

variously called Nullspace homogeneous solarspanof basis solutions

Find one soln to the given problem particular sin

Full Soh is Sun of results from and

Even Ch 5 is in this vein

ay by tey fct glo yo y'Co y

LI ay by tey f fits

al s t sy y bls't y t e't F

as lost c Y f s t asyotay byF Is asyo ay 1byY s as r bas te as'tbste

yit L Hls Ils t L HCskasyotag.i.bg

soIn to soln to

ay bagray o aytbytoy o ya yZero y'Co yICs

partouter homog Solafreedoms disappearedbecauseof Ics

1 I es a ult Ia z sin E

e

i ii f eiIg sin13T

f ult a fit a I e as Lf fits

r

s site product

S s pt

L 198ft ta L IsSinan

8ft sin 3T byConvolutomtheorem

f Sfw Ih Isin 3ft w1 dwe

activatedwhen our_The

O te ith

sin31TALD t Th

comes fromA l f sit I2 f 4 I B't

one slotsys t tc IyD

Ar t B Cs DIz t

is iti1sidesine

Aa ther See individually

L Est t

L l z I 4 since't

Use convolution

I t's I I IsinHEI t w

f t w zsiul2w dwo

fotw.tzs.in2Ct wl dw

3 I l satisturoots of A tbs ta O s 12 I z

ironical1216 11 irreducibleno partialfractures

A S s 3 3s ther t Il s2t6st9 2 s 3 t 2

At 3 Fc 3Gt3 t 2 s 3 t 2 E

I Tcomes from comes from

eatwsll.tl e sin twa 3b iz

Answer I if I e 3tcostrt Ivy e sincere

4 L ye s AA4Gs til

Start L f ft fwm 3

Presence of exponential e m s side

Aas 31T iult l e strut it Zzsinlrzlt.tl