Lets suppose that we know the fundamental system (and it exists) ; we want to find a solution $L[x]=f(t)$.
Let $\forall s \; x(t,s)$ be the solution of a system:
$\left\{ \begin{array}{ll} L[x]=0 \\ x(t_0)=0, x'(t_0)=0, ... , x^{(n-2)}(t_0)=0,x^{(n-1)}(t_0)=1 \end{array} \right.$
then $y(t)=\int_{t_0}^{t} x(t,s)f(s)ds$ is a solution of:
$\left\{ \begin{array}{ll} L[x]=f(t) \\ x^{k}(t_0)=0, k=0,1,...,n-2 \end{array} \right.$
So I have to prove that fact, but don't really know how to proceed. Any help will be much appreciated.