Say that the point x' that y(x') = y'(x') = 0 is x' = 0 for simplicity. Then we know that
(Providing, note, that neither P(x) nor Q(x) are singular at x = 0.) So we know that y(0) = y'(0) = y''(0) = 0. Now take the derivative of the differential equation. This gives you a third degree ODE. Again evaluate the equation at x = 0. This will give y'''(0) = 0. Rinse and repeat.
Now here's that theorem I mentioned: Any function such that for all non-negative integer n is going to have to be identically 0 or if not is going to be discontinuous.
Whether or not I have the theorem exactly correct I think we have to, at some point, stipulate that y(x) is continuous and that P(x) and Q(x) are .
Perhaps that will generate a fruitful approach for you.