How to Find the Inverse Laplace Transform for Ds + E / (s^2 +1)^2?

[kyu]

Homework Statement

Ds + E / (s^2 +1)^2

Homework Equations

The Attempt at a Solution

Ds / (s^2 +1) + E / (s^2 +1)

D[s/(s^2 + 1)^2] + E [1 / (s^2 + 1)^2]

 

[HallsofIvy]

You mean, I assume (Ds+ E)/(s^2+ 1)^2

That can be written as Ds/(s^2+ 1)^2+ E/(s^2+ 1)^2 but you seem to have lost the square on the denominator.

 

[kyu]

i added it.. then what should i do next? no idea how to find the inverse

 

[vela]

Consult a table of Laplace transforms.

 

[Ray Vickson]

i added it.. then what should i do next? no idea how to find the inverse

The standard way is to "know" a number of Laplace transforms already, plus some rules like the connection between the transforms of $f(t)$ and those of $f'(t)$, $\int_0^t f(\tau) \, d\tau$ or $f(t-a)$ for constant $a$--and similar general facts. Then you just try to "recognize" your $\hat{f}(s)$ among those mentioned above, and so know it inverse right away.

There are also general "inversion" formulas, but they are hardly ever used in applications to get inverses.

I suggest you struggle with this problem; it will teach you a lot, and you will be stronger for it after you are finished.