# 2001 S3 Q1

Can I get some help on how to do the proof by induction?

I am able to prove the base case n=0.

I have tried to substitute in the expression the cases n= k+1 and n=k-1, but that leads me nowhere. I have as well unsuccessfully tried to integrate and differentiate the given expression.

I think strong induction may be needed here....

Thank you!

### Actually, there's no need for

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.

### Thank you! It worked

Thank you! It worked perfectly.

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