Obtain the inverse Laplace transformation of $\frac{5s^3-12s^2+11s+2}{s(s-1)^3}$

Question

Solving $\mathcal{L}^{-1}\left\{\dfrac{5s^3-12s^2+11s+2}{s(s-1)^3}\right\}$

I get $\mathcal{L}[f]=3e^tt^2-4e^tt+7e^t-2$, however the options I have are

What am I doing wrong? Is there anything wrong with the question itself?

Answer 1

I have tried by myself and I found that the problem proposition has a mistake. It has to be

$$\mathcal{L}^{-1}\left\{\dfrac{5s^3-12s^2+11s\color{red}{-2}}{s(s-1)^3}\right\}$$

So the correct option will be

$$D.\quad 2+3e^3+t^2e^t$$

Answer 2

Have you used http://mathworld.wolfram.com/PartialFractionDecomposition.html correctly?

$$\dfrac{5s^3-12s^2+11s+3}{s(s-1)^3}$$ $$=\dfrac as+\dfrac b{s-1}+\dfrac c{(s-1)^2}+\dfrac d{(s-1)^3}$$ where $a,b,c,d$ are arbitrary constants to be determined by comparing the coefficients of different powers of $s$