Solving partial fraction expansion with all variables

Question

Okay so I have an equation in my book which is as follows.. $$ \frac {a}{s(s+a)} $$ it says "using partial fractions this can be expanded to $$ \frac {1}{s} + \frac {-1}{s+a} $$

My usual method would be to cross multiply and do something like this $$ \frac {a}{s(s+a)} = \frac {A(s+a)}{s(s+a)} + \frac {B(s)}{s(s+a)} $$

Then cancel off the denominators and solve..

$$ a = A(s+a) + B(s) $$

usually though the a would be some constant but here I have no values to play around with.. how has he done it in the book?

Answer 1

We have:

$$\dfrac{a}{s(s+a)} = \dfrac{A}{s}+\dfrac{B}{s+a}$$

So,

$$a = A(s+a) + Bs = (A+B)s + A a$$

we have $A = 1, B = -1$

Final result:

$$\dfrac{a}{s(s+1)} = \dfrac{1}{s}-\dfrac{1}{s+a}$$

Answer 2

The equality you have is $$ a=(A-B)s+aA. $$ This suggests taking $A=B=1$.

Answer 3

Let $s=0$, so $a=Aa$. Let $s=1$, so $a=A(a+1)+B$
Solve for $A$ and $B$.