Hi guys, I'm new here but I really need some help. I have to compute the inverse Laplace transform of (s-2)/(s^2+4s+5), problem is we barely covered Laplace transforms today and I cannot find a way to rewrite this as a transform that I can find in a table. Can someone help with the partial fraction decomposition?
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Inverse Laplace Transform Question
- Thread starter fruggz
- Start date
Hi guys, I'm new here but I really need some help. I have to compute the inverse Laplace transform of (s-2)/(s^2+4s+5), problem is we barely covered Laplace transforms today and I cannot find a way to rewrite this as a transform that I can find in a table. Can someone help with the partial fraction decomposition?
If you just want the partial fraction decomposition, let s0 and s1 be roots to s2 + 4s + 5
and write
\(\displaystyle \frac{s-2}{s^2+4s+5}=\frac{As+B}{s-s_0} + \frac{Cs+D}{s-s_1}\)
since
(s-s0) (s-s1) = s2 + 4x + 5
s0 + s1 = -4
s0 s1 = 5
Now work out the A, B, C, D. That is
\(\displaystyle \frac{s-2}{s^2+4s+5}=\frac{(As+B)(s-s_1)+(Cs+D)(s-s_0)} {s^2+4s+5}\)
Last edited:
HallsofIvy
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\(\displaystyle s^2+ 4x+ 5= s^2+ 4x+ 4- 4+ 5= (s+ 2)^2+ 1\)
That cannot be factored into "\(\displaystyle (s- s_0)(s- s_1)\)" with \(\displaystyle s_0\) and \(\displaystyle s_1\) real numbers so you should not use partial fractions. Instead, write \(\displaystyle \frac{s- 2}{s^2+ 4x+ 5}\) as \(\displaystyle \frac{s}{(s+ 2)^2+ 1}- 2\frac{1}{(s+2)^2+ 1}\) and look up the inverse transform for each of those.
That cannot be factored into "\(\displaystyle (s- s_0)(s- s_1)\)" with \(\displaystyle s_0\) and \(\displaystyle s_1\) real numbers so you should not use partial fractions. Instead, write \(\displaystyle \frac{s- 2}{s^2+ 4x+ 5}\) as \(\displaystyle \frac{s}{(s+ 2)^2+ 1}- 2\frac{1}{(s+2)^2+ 1}\) and look up the inverse transform for each of those.
\(\displaystyle s^2+ 4x+ 5= s^2+ 4x+ 4- 4+ 5= (s+ 2)^2+ 1\)
That cannot be factored into "\(\displaystyle (s- s_0)(s- s_1)\)" with \(\displaystyle s_0\) and \(\displaystyle s_1\) real numbers so you should not use partial fractions. Instead, write \(\displaystyle \frac{s- 2}{s^2+ 4x+ 5}\) as \(\displaystyle \frac{s}{(s+ 2)^2+ 1}- 2\frac{1}{(s+2)^2+ 1}\) and look up the inverse transform for each of those.
It might be easier if you try
\(\displaystyle \frac{s- 2}{s^2+ 4x+ 5}\) as \(\displaystyle \frac{s+2}{(s+2)^2+ 1}- 4\frac{1}{(s+2)^2+ 1}\)