The 'conventional' approach to a linear constant coefficients ODE...

$\displaystyle y^{'} + a\ y = g(t)$ (1)

... works very well when $\displaystyle g(t)$ is a

*continous function*. In Your case however $\displaystyle g(t)$ contains the Heaviside step function...

$\displaystyle \mathcal{U}(t)=\begin{cases}0&t<0\\ \frac{1}{2}&t=0\\ 1&t>0\end{cases}$ (2)

... which is not continous. Oliver Heaviside was a self-taught English electrical engineer [as I am...], born in 1850, who invented, among other 'wonders', mathematical techniques to the solution of differential equations that at the time were considered 'unsolvable'. But Heaviside wasn't a mathematician and above all was self-taugh, so that he was in any way 'defamed' by the 'Holy Accademy' till to be expelled from the Royal Society for 'unworthiness'

...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$