Linearity: The Laplace transform is a linear operation L(ax(t) + by(t)) = aX(s) + bY(s) Integral: the Laplace transform of a function that is integrated with respect to time x { } = = f(t*) dt* S ► Multiplication by time: F(s) L{tf(t)} = -F'(s) = F(s) ds 11

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Prove all eight properties of the Laplace transform

Linearity: The Laplace transform is a linear operation
L(ax(t) + by(t)) = aX(s) + bY(s)
Integral: the Laplace transform of a function that is integrated with respect
to time
L
{"
Multiplication by time:
1
f(t*) dt* = F(s)
S
d
L{tf(t)} = -F' (s) == F(s)
−F'
ds
11
Transcribed Image Text:Linearity: The Laplace transform is a linear operation L(ax(t) + by(t)) = aX(s) + bY(s) Integral: the Laplace transform of a function that is integrated with respect to time L {" Multiplication by time: 1 f(t*) dt* = F(s) S d L{tf(t)} = -F' (s) == F(s) −F' ds 11
► Change of Scale:
► Complex Shift:
Time Shift:
Initial Value:
► Final Value:
1
L {f(at)} = = F ( ² )
a
α
L{eat f(t)} = F(s – a)
-
L(f(t – to)S(t — to)) = e¯stoF(s)
lim y(t) = lim [sY(s)]
t→0
$→∞
lim y(t) = lim [sY(s)]
t→∞
S-0
Transcribed Image Text:► Change of Scale: ► Complex Shift: Time Shift: Initial Value: ► Final Value: 1 L {f(at)} = = F ( ² ) a α L{eat f(t)} = F(s – a) - L(f(t – to)S(t — to)) = e¯stoF(s) lim y(t) = lim [sY(s)] t→0 $→∞ lim y(t) = lim [sY(s)] t→∞ S-0
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