WTW256: LU 3.1: LAPLACE TRANSFORMS AND INVERSE TRANSFORMS Lecture notes

3. 1 . Laplace transform
↳ another method to solve linear bf 's with
constant coeff

Use④
.




when RHS of bf is a piecewise continuous function


{
t ◦≤ tis
get
-
-




4 t≥ 3

Steps

1. say ylt) is the
general solution of bf
T yet)= ? Apply tdbld "
> Algebraic equation is s
t to every term
^ in DE




2-
i write "eq 's with L{y }
F' { Ily } } <
Apply inverse
9s subject
laplace transf L{j}=
yet)
. . .
.

= .




Revision Improper
:
integrals
J ? fltldt converges if
tim
b. →• f? fctdt exists and then

I ?HHdt= t.it?,J!tctidt
Laplace
Lett be defined for t≥ 0 _ The laplace transform

off , L{ fit)} ,
is defined as
J?e-stfctsdt if the
integral converges coexists)
-




[ L { f) -
-




JIE
_
"
fagot ]

, •

Notation -
- -
-

- ,


small letter function
iL{ t} FCS ) '
→ :
=
'




→ capital letter :
Laplace \
- -
-
-
-
-




LAPLACE TRANSFORM
Examples
1. use definition of laplace transform to find

I. { 3 } ""
g) ^ fits
'
bi:S ! e-stat
' 3
consider


ru >


f. e-stfltdt]
Need to find out if it converges
integrate t is the
w r t Iim -

SE →
Se
- .
-
=

variable
④ by -
s

Iim sbb
geico)
=




( )
3C
-




b. →•
-




-

s -


s
=
Iim
i'
'

se sbb → Doesn't have
-




7%1
-




b→•
"



#
e
' tends to

a' b in . so

g
we have
only limit
↓ to cdlc .




exponential we want it
> ᵗes
function to look like zero
this graph
[ needs to be:] :b has to be positive
④
needs
_g④
:




b⊕Ñ
power
to be negative
Zero

→ [ start { if
-




ilim jeisb '
3C s> o
-
=


=3 ,
o
b. →0 "
s> LE} } -31s s>
'




n-o-j.it 0 o
-



_
: .




i
-