I would like to ask for some hint to the following problem :
Let $y(t,\alpha)$ be the solution of the system :
$$ \left\{ \begin{array}{ll} Lx=0 \\ x(\alpha)=x'(\alpha)=\cdots=x^{(n-1)}(\alpha)=0, x^{(n)}(\alpha)=1 \end{array} \right.$$
Prove that $\phi $(t) = $\int_{t_0}^t y(t,\alpha) f(\alpha) \, d\alpha $ is the solution of :
$$ \left\{ \begin{array}{ll} Lx=f \\ \phi(t_0)=\phi'(t_0)=\cdots=\phi^{(n-1)}(t_0)=0 \end{array} \right.$$