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Mutaja
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Homework Statement
Find the inverse Laplace transform of the expression:
F(S) = [itex]\frac{3s+5}{s^2 +9}[/itex]
Homework Equations
The Attempt at a Solution
From general Laplace transforms, I see a pattern with laplace transforming sin(t) and cos(t) because:
L{sin(t)+cos(t)} = [itex]\frac{s+1}{s^2 +1}[/itex]
All I'm missing here is a couple of constants(?).
I know that the laplace transform works like this:
L{Asin(Bt)+Ccos(Dt)} = [itex]\frac{Cs}{s^2 +D}[/itex] + [itex]\frac{A*B}{s^2 +B^2}[/itex]
Looking at my original problem, I can see that I need B to equal D, ##D^2## + ##B^2## = 9, C = 3 and A*B = 5.
If I set A = 5, B = 3, C = 9 and D = 3 I get ##3^2## + ##3^2## = 18, 5
3 = 15 (putting numbers into my equations above).
Therefore I need to divide the whole expression by 3 to get my correct answer:
L{[itex]\frac{1}{3}([/itex]5sin(3t)+9cos(3t))} = [itex]\frac{3s+5}{s^2 +9}[/itex]
Is there a simpler way to do this? Or do I just have to use the trial and error method until I find the correct factors/constants?
Any input on this will be greatly appreciated. Thanks