Is $\cos(x^2)$ a solution of a 2nd order linear ODE defined on all of $\mathbb R$? I've been asked in an assignment whether the function $\cos(x^2)$ can be the solution of a 2nd order homogeneous linear equation which satisfies the existence and uniqueness theorem on all of $\mathbb R$. On one hand I cannot find a reason why it wouldn't be a solution of such an equation (unlike $\sin(x^2)$, which was easy) since there is no point where both the function and its derivative vanish, on the other hand I cannot for the life of me find an example of such an equation. Assistance would be appreciated.

Your function is a solution of $$xy''-y'+4x^3y=0.$$
