Laplace Transformation using Heaviside functions

Question

I'm not very familiar with Heaviside functions so I am struggling with this:

I'm supposed to compute $Lu$ where $u''+4u=H(x-0)+H(x-\pi)$ and $H$ is a Heaviside function. Any suggestions are greatly appreciated!

Answer 1

We should start with the definition. $$ \mathcal{L}\{f(t)\} = \int_0^{\infty}f(t)e^{-st}dt $$ Then we can compute the Laplace transform as \begin{align} \mathcal{L}(u'') + 4\mathcal{L}(u) &= \mathcal{L}\{H(x)\} + \mathcal{L}\{H(x - \pi)\}\\ s^2U(s) - su(0) - u'(0) + 4U(s) &= \int_0^{\infty}e^{-st}dt + \int_{\pi}^{\infty}e^{-st}dt \end{align}