A solution of a given ODE is infinitely differentiable

Question

I got this problem in a class test:

Prove that any solution of the DE $$y''e^x+y'\cos{x}+e^{2x^2}y=0$$ is infinitely differentiable.

I solved this as follows but I'm not entirely sure if I'm correct:

Let $y_0$ be a solution of the given DE. I need to prove that $y_0^{(n)}$ exists $\forall$ $n\in \mathbb{N}$. I prove this by induction.

The statement is true for $n=1,2$ because the order of the DE is $2$.

Induction hypothesis: Let the statement be true $\forall$ $n

Now, $y_0''e^x= -y_0'\cos{x} -e^{2x^2}y_0 \Rightarrow y_0'' = F(y_0',y_0,x)$. Differentiating this $k-3$ times, I have $$y_0^{(k-1)}=G(y_0^{(k-2)}, \cdots, y_0', y_0, x)$$ By the induction hypothesis, the right hand side is differentiable, implying that $y_0^{(k)}$ exists.

Is my solution correct? If not, how do I solve this problem?