- #1
JJBladester
Gold Member
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Homework Statement
Find [tex]f(t)[/tex].
Homework Equations
[tex]L^{-1}\left\{\frac{s}{s^{2}+4s+5}\right\}[/tex]
The Attempt at a Solution
I tried completing the square to get to the solution and I ended up with:
[tex]L^{-1}\left\{\frac{s}{s^{2}+4s+5}\right\}[/tex] =
[tex]L^{-1}\left\{\frac{s}{(s+2)^{2}+1}\right\}[/tex]
Then, I used the inverse of a transform for cosine and the first translation theorum:
[tex]coskt = L^{-1}\left\{\frac{s}{(s^{2}+k^{2})}\right\}[/tex]
[tex]L\left\{e^{at}f(t)\right\} = F(s-a)}[/tex]
with [tex]a[/tex] being -2 and [tex]k[/tex] being 1 to get an answer of:
[tex]e^{-2t}cos(t)[/tex]
However, I was wrong. The book had the following answer:
[tex]e^{-2t}cos(t)-2e^{-2t}sin(t)[/tex]
My question is where does the [tex]-2e^{-2t}sin(t)[/tex] come from?