Suppose I have a fraction as
$\frac{1}{s}\frac{1}{s^2-2s+5}$
So that:
$1 = (A+B)s^2+(C-2A)s+5A$
I'm just confused as to how I can solve for $A,B,$ and $C$. I know if we plug $s=0$ we get $A = \frac{1}{5}$, but what can I do for B and C?
Suppose I have a fraction as
$\frac{1}{s}\frac{1}{s^2-2s+5}$
So that:
$1 = (A+B)s^2+(C-2A)s+5A$
I'm just confused as to how I can solve for $A,B,$ and $C$. I know if we plug $s=0$ we get $A = \frac{1}{5}$, but what can I do for B and C?
Both sides of the equation are polynomials in $s$ (the LHS just happens to be a constant polynomial), so you can compare coefficients of $s^2$ on both sides. The coefficient of $s^2$ is zero on LHS and $A+B$ on RHS. So $A+B=0$. Do the similar for the coefficient of $s$.
You have $A + B = 0$ and $C - 2A = 0$. Use 'em.