Laplace transform of a non-homogeneous differential equation

Question

I don't understand how to to write the answer of this problem in in the form $$ A\frac {1}{s-1}+ B\frac {1}{s+1} + C\frac {1}{s+2} $$ depicted in this problem:

Answer 1

You have \begin{align*} \mathcal{L}\big\{5y'''+11y''-5y'-11y&=e^{-2t}\big\}\\ 5\left[s^3 Y(s)-s^2 y(0)-s y'(0)-y''(0)\right]+11\left[s^2 Y(s)-s y(0)-y'(0)\right] -5\left[sY(s)-y(0)\right]-11Y(s)&=\frac{1}{s+2}. \end{align*} Continuing: \begin{align*} 5\left[s^3 Y(s)-1\right]+11s^2 Y(s)-5sY(s)-11Y(s)&=\frac{1}{s+2}\\ Y(s)\left[5s^3+11s^2-5s-11\right]&=\frac{1}{s+2}+5\frac{s+2}{s+2}\\ Y(s)\left[5s^3+11s^2-5s-11\right]&=\frac{5s+11}{s+2}\\ Y(s)&=\frac{5s+11}{(s+2)(5s^3+11s^2-5s-11)}\\ &=\frac{5s+11}{(s+2)(s-1)(s+1)(5s+11)}\\ &=\frac{1}{(s+2)(s-1)(s+1)},\quad\text{for}\; s\not=-11/5. \end{align*} Next, you assume, via partial fractions, that \begin{align*} \frac{1}{(s+2)(s-1)(s+1)} &=\frac{A}{s-1}+\frac{B}{s+1}+\frac{C}{s+2}. \end{align*} The most straight-forward way to get $A,B,$ and $C$ is to get the common denominators on the RHS, and equate the same powers of $s,$ thus giving you three equations in three unknowns. However, in this case, the Heaviside cover-up method will give you the results much faster. We have, in fact, that \begin{align*} A&=1/6\\ B&=-1/2\\ C&=1/3. \end{align*}