Linear differential equation/Solution/Differentiability/0/All derivatives 0/Exercise

Let

${}y'=g(t)y\,$

be a homogeneous linear ordinary differential equation with a function ${}g$ differentiable infinitely many times and let ${}y$ be a differentiable solution.

a) Prove that ${}y$ is also infinitely differentiable.

b) Let ${}y(t_{0})=0$ for a time-point ${}t_{0}$. Prove, using the formula

${}(f\cdot g)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}f^{(k)}\cdot g^{(n-k)}\,,$

that ${}y^{(n)}(t_{0})=0$ for all ${}n\in \mathbb {N}$.

Create a solution