Trying to solve the follow equation:
$$y^{\prime \prime} + y = \begin{cases}
t^2 & 0\leq t \leq 1, \\
0 & \text{else}
\end{cases}$$
with the initial conditions $y(0) = y^\prime(0) = 0$.
I get an expresion for the laplace transfor of $y$:
$$L(y) = \frac{1-e^{-s}}{s(s^2 +1)} + \frac{1-e^{-s}}{s^2(s^2 +1)} + \frac{1-e^{-s}}{s^3(s^2 +1)}.$$
I can separate each one and use partial fractions, but since all expressions are similar, is there an easier way to solve this? I ask because I have no idea how to take the inverse laplace transform of $\frac{e^{-s}}{s^n(s^2 +1)}.$
As you know, $L^{-1}(e^{-s} F(s))=f(t-1) H(t-1)$ where $H$ is the Heaviside function. So your question is really just about how to deal with $\frac{1}{s^n(s^2+1)}$. If you don't want to actually do the partial fractions then you have to deal with convolutions instead, i.e. you have $\int_0^t f(s) g(t-s) ds$ where $f$ is the inverse Laplace transform of $\frac{1}{s^n}$ (which is proportional to $t^{n-1}$) and $g$ is the inverse Laplace transform of $\frac{1}{s^2+1}$ which is of course $\sin(t)$. This is probably harder than just doing the partial fractions.