- #1
Melawrghk
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Homework Statement
Find the solution of the given initial value problem:
y''+4y=upi(t)-u3pi(t) y(0)=7, y'(0)=5
The Attempt at a Solution
So I found the L{} of the above equation:
s2Y-s*f(0)-f'(0)+4Y = (e-pi*s)/s-(e-3pi*s)/s
Combining and substituting the numbers I get:
Y=[tex]\frac{e^{-pi*s}-e^{-3pi*s}}{s(s^{2}+4)}[/tex]+[tex]\frac{6s+3}{s^2+4}[/tex]
I know how to do the second term's inverse Laplace, but not the first. Here is what I tried:
I can see that I can't get rid of the exponentials in any way other than using the step function again. And the other denominator factor (s^2+4) can be potentially used to get sine. So that:
Y=[tex]\frac{1}{2}(\frac{e^{-pi*s}-e^{-3pi*s}}{s})[/tex][tex]\frac{2}{s^{2}+4}[/tex]
This is where I don't know what to do. I can't separate them and I don't know of a way to do Laplace inverse of a product.
Any help would be really appreciated. Thanks
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