How Do You Solve Complex Inverse Laplace Transforms?

[jegues]

Homework Statement

Find the inverse Laplace Transform for the function,

$$F(s) = \frac{(3s^{2}+9s+14)e^{-5s}}{s^3 + 4s^2 +7s}$$

Homework Equations

The Attempt at a Solution

We are given a table of common laplace transforms that we are allowed use to without proof.

First I'll rewrite F(s) like so,

$$\frac{(3s^{2}+9s+14)}{s^3 + 4s^2 +7s} \cdot e^{-5s}$$

because all the exponential term is doing is the 2nd translation theorem which I'll deal with near the end, so for now I'm focusing on term on the left hand side.

I've been trying to factorize the top and bottom and make some cancelations or possibly complete the square but I can't come up with anything pretty.

Does anyone have any suggestions? I'm seeing the obvious stuff like we can pull out and s on the bottom, but I don't see what factorizations I should make.

 

[Mark44]

Homework Statement

Find the inverse Laplace Transform for the function,

$$F(s) = \frac{(3s^{2}+9s+14)e^{-5s}}{s^3 + 4s^2 +7s}$$

Homework Equations

The Attempt at a Solution

We are given a table of common laplace transforms that we are allowed use to without proof.

First I'll rewrite F(s) like so,

$$\frac{(3s^{2}+9s+14)}{s^3 + 4s^2 +7s} \cdot e^{-5s}$$

because all the exponential term is doing is the 2nd translation theorem which I'll deal with near the end, so for now I'm focusing on term on the left hand side.

I've been trying to factorize the top and bottom and make some cancelations or possibly complete the square but I can't come up with anything pretty.

Does anyone have any suggestions? I'm seeing the obvious stuff like we can pull out and s on the bottom, but I don't see what factorizations I should make.

Factor the denominator into s(s² + 4s + 7) and then use partial fractions. The will be of the form A/s + (Bs + C)/(s² + 4s + 7).