Is this the correct setup for partial fractions? $\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2}$

Question

I am trying to inverse laplace transform the following:

$$F(s)=\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}$$

and I believe what I do is take:

$$\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2}$$

and then I get:

$$1-e^{-s} + se^{-s} + s^3=A*s(s^2+2)+B(s^2+2)+(C*s+D)s^2$$

Is this the correct setup?

Note: I know what to do after this, if it is the correct setup.

Answer 1

Yes, it is. Treat $s$ like $(s+0)$, so it becomes a repeated linear factor.

For $$\frac{f(s)}{s^2(s^2+2)}=\frac{f(s)}{(s+0)^2(s^2+2)}$$

Then you use the rule that $$\frac{f(s)}{(s+a)^2(s^2+b)}=\frac{A}{s+0}+\frac{B}{(s+0)^2}+\frac{Cx+D}{s^2+b}$$