Actually, there's no need for strong induction. You've assumed $$(x^2 + 1)y^{(n+2)} + (2n+1)xy^{(n+1)} + n^2y^{(n)} = 0.$$

Now, this is quite a common STEP-style question and the immediate thing to do is to differentiate the above with respect to $x$. The first term becomes $\frac{d}{dx}((x^2+1)y^{(n+2)}) = \frac{d}{dx}(x^2+1)y^{(n+2)} + (x^2+1)\frac{d}{dx}y^{(n+2)} = 2xy^{(n+2)} + (x^2+1)y^{(n+3)}$ via the product rule.

Can you do something similar for $\frac{d}{dx}((2n+1)xy^{(n+1)})$ and $\frac{d}{dx}(n^2 y^{(n)}$? Once you have, put everything together, collect terms and you end up with the thing you want to show inductively! :-)

Let me know if this isn't clear enough.