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WTW256: LU 3.1: LAPLACE TRANSFORMS AND INVERSE TRANSFORMS Lecture notes $4.57   Add to cart

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

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Lecture notes were made while watching the recorded lectures assigned to watch. These notes include theory (theorems) and worked out examples from the lecturer. These specific notes cover Laplace transforms and inverse Laplace transforms.

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  • July 11, 2022
  • 7
  • 2021/2022
  • Class notes
  • Ms l mostert
  • All classes
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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

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
-

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