Solving differential equation involving Heaviside's unit step function using Laplace Transform

Question

$\ddot y(t)+2\dot y(t)+2y(t)=5H(t-2\pi)sint$

Also, $y(0)=1$ and $\dot y(0)=0$

I took the usual approach of solving an initial value problem but things get a bit tricky after the first few steps due to involvement of Heaviside's function.

Any help is appreciated.

Answer 1

Hint:) ‎\begin{eqnarray*}‎ ‎&& {\cal L}(y^{\prime\prime})+2{\cal L}(y^{\prime})+2{\cal L}(y)={\cal L}(5H(t-2\pi)\sin t) \\‎ ‎&& s^2{\cal L}(y)-sy(0)-y^\prime(0)+2\Big(s{\cal L}(y)-y(0)\Big)+2{\cal L}(y)=5\dfrac{e^{-2\pi s}}{s^2+1} \\‎ ‎&& {\cal L}(y)\Big(s^2+2s+2\Big)=5\dfrac{e^{-2\pi s}}{s^2+1}+s+2 \\‎ ‎&& {\cal L}(y)=5\dfrac{e^{-2\pi s}}{(s^2+1)(s^2+2s+2)}+\frac{s+2}{s^2+2s+2} ‎\end{eqnarray*}‎ Now using partial-fraction decomposition continue!

Answer 2

Try using

$$\mathcal{L}\{H(t-a)g(t)\}=\mathcal{L}\{g(t+a)\}e^{-as}$$