When does a differential equation
$\frac{d^2y}{dt^2}+a\frac{dy}{dt}+by=cu(t)+d\frac{du}{dt}$
admit a solution? If $d=0$, the existence is answered by Picard-lindelöf, and we can write it as a system on the form
$\frac{dx}{dt}=Ax(t)+Bu(t)$
where $x$ is a vector by letting $x_1=y$, $x_2=\frac{dy}{dt}$ and so on. When can we do that for the equation above with $d \neq 0$, and how?