Can someone give me hint on how to show that the fact that $\alpha$ is a root of order $l+1$ of the polynomial $p(x)=x^n+a_1x^{n-1}+\cdots+a_n$ implies that $t\mapsto y^l e^{\lambda y}$ is a solution of the ODE $x^{(n)}+a_1 x^{(n-1)}+\cdots + a_n x=0$ ?
I tried doing a direct calculation, but didn't succeed (altough I would prefer this direct route, since I don't yet see all the connection between the different theorems of the theory for linear systems of ODEs, so it would be easier for me to understand)...then I tried using some theory about ODEs but didn't get far: The ODE is equivalent to a system of linear ODEs of order 1 and the characteristic polynomial of the matrix of that system is precisely the polynomial above, but there I kind of stop dead, since calculating the Jordan normal form is just horrible, so I thought there must be some easier, less calculation-heavy way. Could you show me the way ?