The following ODE is given: $$y''(t) + p(t)y'(t) + q(t)y(t)=0$$
When $p(t), q(t)$ are continuous functions. We are given two linear independent solutions $y_1(t), y_2(t)$ and also $y_1''(t_0) = y_2''(t_0) = 0$.
I need to prove that $p(t_0) = q(t_0) = 0$.
What I've tried is just placing zero in the second derivative for each function in the ODE, and working with the Wronskian. However I end up with $$p(t)(y_1'(t_0) - y_2'(t_0)) + q(t)(y_1(t_0) - y_2(t_0))$$ which is not the Wronskian.