2023 SJI P1 Q9

Consider \(\ln y=x^2\).

1. Show that \(\dfrac{\mathrm dy}{\mathrm dx}=2xy\).[2]
2. 1. Find an expression for \(\dfrac{\mathrm d^2y}{\mathrm dx^2}\), giving your answer in terms of \(x\), \(y\) and \(\dfrac{\mathrm dy}{\mathrm dx}\).
   2. Find the value of \(\dfrac{\mathrm d^2y}{\mathrm dx^2}\) when \(x=0\).[4]
3. 1. By further differentiation, show that \(\dfrac{\mathrm d^3y}{\mathrm dx^3}=2x\dfrac{\mathrm d^2y}{\mathrm dx^2}+4\dfrac{\mathrm dy}{\mathrm dx}\).
   2. Hence, find \(\dfrac{\mathrm d^4y}{\mathrm dx^4}\) in a similar form.[4]
4. Given that \(\dfrac{\mathrm d^ny}{\mathrm dx^n}=rx\dfrac{\mathrm d^{n-1}y}{\mathrm dx^{n-1}}+s\dfrac{\mathrm d^{n-2}y}{\mathrm dx^{n-2}}\), where \(n\in\mathbb Z\), \(n\geq3\), deduce the value of \(r\) and the expression of \(s\) in terms of \(n\).[2]
5. Suppose \(D_n\), where \(n\in\mathbb Z^+\), denotes the value of \(\dfrac{\mathrm d^ny}{\mathrm dx^n}\) when \(x=0\). Show that
   1. \(D_n=0\) when \(n\) is odd;
   2. \(D_n={}^n\mathrm P_{n/2}\) when \(n\) is even.[5]