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WTW256: LU 3.2: TRANSFORMATION OF INITIAL VALUE PROBLEMS Lecture notes $4.67   Add to cart
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WTW256: LU 3.2: TRANSFORMATION OF INITIAL VALUE PROBLEMS Lecture notes
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  • Course
  • WTW 256 - Differential Equations
  • Institution
  • University Of Pretoria (UP)

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 Transformation of initial value problems.

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Document information

  • July 11, 2022
  • 6
  • 2021/2022
  • Class notes
  • Ms l mostert
  • All classes

Subjects

  • wtw
  • wtw256
  • 256
  • laplace
  • laplace transform

Written for

  • University of Pretoria (UP)
  • WTW 256 - Differential Equations
All documents for this subject (43)

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3. 2. Transformation of / UP

* when is continuous function but a
glx ) not a

piece wise function
-

.




Derivatives of Laplace transforms
first derivative
L { 1-41-1 }=gf( s) -

f / a)

L { ] S2 Fcs) s f- (e) f / c)
'

f- ( t)
-
" -
- -




second derivative

I { eat fits } =


Fcs -




d)
I{ fit d) -



yet a) } - =
e-
as
Fcs)

L{ get yet a) ] -
-

-




e-
"
L { gotra ) }


Theorem 7.2.2 .
P2 89

I{ f' " } snL{ f }
=
-

Sn flat
-

' -


Sn
-
'
f
'

lo ) -


. . .
f-
'" "
(a)

If A- 1 PROOF


L{ f / SL { f)
'
= -




fed
* Need to be able to prove !*
'

① Apply definition of Laplace transform -


to t that is

L{ f) =


5. e-" 1- '


( Hot

② use integration by parts
③ To fÑÉÉÑthe fact that tis of exp order .
t
Squee • a
Tthheeoreemm

, lim b
/
"
b →8 e- t ( Hdt
a




f!
"
d) e- f 't Hdt
b Int




st-fitijbj.es/obfitse-stdl:ie-i b- i b-ji -i. i.s :titie-stdt
=




- -
- -
- -
-




b) Iim
☆→ • (e- ¥1b ) -
f (a) asf:b fit )e stdt )
-




limb
=




→ • e-sbflb )
-


flat is L{ f)

/ e-sbfcb ) / / e-sb / / fib) / e-sbmecb
≤
◦ ≤ =
b≥ 7

proof continued in appendix

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