Im having some trouble when I apply partial fraction decomposition on $$\frac{s^{2}+1}{s^{3}-s} $$ It can be simplified to $\frac{s^{2}+1}{s(s-1)(s+1)}$. Isn't the aim after this step to rewrite the expression into $$\frac{s^{2}+1}{s(s-1)(s+1)}=\frac{A}{s}+\frac{B}{s-1}+\frac{C}{s+1} $$?
It doesnt seem to work when I find the values for $A,B,C$. Is there something I have missed that maybe should be included in the numerators or?
On
You get by identifying coefficients for each monomial:$$\cases{\phantom{-}A+B+C=1\\\phantom{-A}+B-C=0\\-A\phantom{+B-C}\,\,\,=1}$$
This is a linear equation system we can write on matrix form:
$$\left[\begin{array}{rrr|r}1&1&1&1\\0&1&-1&0\\-1&0&0&1\end{array}\right]$$
If we solve it we see we get $[-1,1,1]^T$ corresponding to $A=-1,B=C=1$
$$\frac{s^{2}+1}{s(s-1)(s+1)}=\frac{A}{s}+\frac{B}{s-1}+\frac{C}{s+1}$$
$$\implies s^2 +1 = A(s^2 -1) + B(s^2+s) + C(s^2 - s)$$
Letting $s = 1$ gives $B=1$, letting $s = -1$ then gives $C =1 $, and we then substitute to get $A = -1$.