Inverse Laplace Transform of this expression?

[supermiedos]

Homework Statement

Find the Inverse Laplace Transform of $\frac{1}{(s^{2} + 1)^{2}}$

Homework Equations

The Attempt at a Solution

I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(

 

[LCKurtz]

Homework Statement

Find the Inverse Laplace Transform of $\frac{1}{(s^{2} + 1)^{2}}$

Homework Equations

The Attempt at a Solution

I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(

Think of it as$$
\frac 1 {s^2+1}\cdot \frac 1 {s^2+1}$$What theorem do you know about the product of transforms?

 

[supermiedos]

Ohh the convolution theorem?? I'm working on it

 

[supermiedos]

I got it now. Thank you so much

 

[Ray Vickson]

Homework Statement

Find the Inverse Laplace Transform of $\frac{1}{(s^{2} + 1)^{2}}$

Homework Equations

The Attempt at a Solution

I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(

Another way: start with
$$g_a(s) = \frac{1}{(s^2+1)(s^2+a^2)}, \: a \neq 1$$
expand in partial fractions, find the inverse Laplace transform $F_a(t)$, then take the limit as $a \to 1.$

 

[supermiedos]

I expanded as you suggested and applied the transform. I got:

$\frac{sin t}{a^{2} - 1}$ + $\frac{sin(at)}{a(1 - a^{2} )}$

But if I try to take the limit as a goes to 1, they just go to infinity. What am I doing wrong?

 

[Ray Vickson]

I expanded as you suggested and applied the transform. I got:

$\frac{sin t}{a^{2} - 1}$ + $\frac{sin(at)}{a(1 - a^{2} )}$

But if I try to take the limit as a goes to 1, they just go to infinity. What am I doing wrong?

Set $a = 1+\epsilon$, and expand things out until you get something in which you can use l'Hospital's rule to evaluate the limit as $\epsilon \to 0.$ I've done it, and it works!

Or, you can write your result as
$$\frac{\sin(at) - a \sin(t)}{a(1-a^2)}$$ and then use l'Hospital.

 

[supermiedos]

Omg you are right. I made the sum of fractions and used L'Hopital rule... I got the same result. That was amazing! I learned a valuable method today, thank you