# Find the given Laplace Transform

I thought that I had almost figured this problem out. Here is my problem:

$\mathcal {L}^{-1} \left\{ \frac{1}{(s^2+1)(s^2+25)} \right\}$

The final answer that I had got was $\frac{1}{24 sin(t) -\frac{1}{120} cos(5t)}$, but was incorrect. I feel like I had maybe left something off towards the end. Any help is very much appreciated.

• Use partial fractions Commented Nov 6, 2017 at 1:12
• Certainly the wanted inverse Laplace transform is an entire function, so it cannot be something like $\frac{1}{24\sin t-\frac{1}{120}\cos(5t)}$, which has a lot of poles. Commented Nov 6, 2017 at 1:22
• How did you get the answer you did? On learning's answer, you commented that you mixed up sin and cos. I'm more concerned that you ended up with sin and cos in the denominator. Commented Nov 6, 2017 at 3:39

$$\frac{1}{(s^2+1)(s^2+25)} = \frac{1}{24}\left(\frac{1}{s^2+1}-\frac{1}{s^2+25}\right)$$ and since $\mathcal{L}^{-1}\left(\frac{1}{1+s^2}\right)=\sin x$, the inverse Laplace transform of the given function is $$\frac{1}{24}\left(\sin x-\frac{1}{5}\sin(5x)\right).$$

$$F(s) = \frac{1}{(s^2+1)(s^2 + 25)}$$

Unravel this using partial fractions: $$F(s) = \frac{As+B}{s^2+1} + \frac{Cs+D}{s^2+25} = \frac{As}{s^2+1}+\frac{B}{s^2+1}+\frac{Cs}{s^2+25}+\frac{D}{s^2+25}$$

Calculate$A, B, C, D$ in that step.

Can you use the Inverse Laplace Transform tables to solve it beyond this?

• I did get to that point. I was left with: $s^3(A+C)+s^2(B+D)+s(25A+C)+25B+D=1$ Commented Nov 6, 2017 at 1:18
• $A$ and $C$ are obviously zero since $F(s)$ is an even function. Commented Nov 6, 2017 at 1:20
• @JackD'Aurizio That's perfect (and we don't really need Partial Fractions to see this, it's just number shifting in this particular case). Commented Nov 6, 2017 at 1:21
• @J.Armstrong So once you set the values, you will be left with a sum of two terms. And the Inverse Laplace Transform is linear. So the answer will be a sum of two terms. Commented Nov 6, 2017 at 1:27
• I see what I did now. I had my sin and cos mixed up. Thank you everyone so much for your help. Commented Nov 6, 2017 at 1:34