Let $P$ be a polynomial of complex coefficients, and denote the differentiability operator as $D$.
In this notation, we want to find a basis of solutions of the equation $P(D)u = 0$ continuous in an interval $[a, b]$.
I have arrived to the fact that if $P(t) = (t-\lambda_1)^{m_1}\dots (t-\lambda_q)^{m_q}$ then we have that $t^i e^{\lambda_j t}$ is a solution for $1\le j \le q$, $0\le i < m_j$.
What is left is to show that these solutions are linearly independent. Can anybody give me a simple proof?