Solving for $x$ in a Laplace equation

Question

So I have this Laplace equation:

$$s^{2}x+4sx+5=\frac{s}{s-1}$$

And I want to solve for $x$. My result is the following:

$$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$

Which is also the same answer that for example WolframAlpha gives me. Turns out that this is kind of complicated to do partial fraction decomposition on to thereafter do inverse-Laplace on.

Now in the step-by-step solution for the whole problem (in my book) a nice solve for $x$ is stated, as the following:

$$x = \frac{s}{(s-1)(s^{2}+4s+5)}$$

So my question is: how do I go from the original Laplace equation to my books solution on how to solve $x$?

Answer 1

Bring over the $5$ to the left and factor out $x$: $x(s^2+4s)=\frac{s}{s-1}-5$ Then combine terms on the right to get $x(s^2+4s)=\frac{s-5(s-1)}{s-1}$ Divide both sides by $s^2+4s$ and I come to $x=\frac{-4s+5}{(s-1)(s^2+4s)}$ which is different from what your book stated. In the event of doing PFD, you now have three denom's being $s$ , $s+4$ and $s-1$ with numerators $A,B,C$ This should be standard now. You may share with us your partial fractions. (Give it a try)