# Paper 2, Section II, A

(a) Consider the differential equation

$a_{n} \frac{d^{n} y}{d x^{n}}+a_{n-1} \frac{d^{n-1} y}{d x^{n-1}}+\ldots+a_{2} \frac{d^{2} y}{d x^{2}}+a_{1} \frac{d y}{d x}+a_{0} y=0$

with $n \in \mathbb{N}$ and $a_{0}, \ldots, a_{n} \in \mathbb{R}$. Show that $y(x)=e^{\lambda x}$ is a solution if and only if $p(\lambda)=0$ where

$p(\lambda)=a_{n} \lambda^{n}+a_{n-1} \lambda^{n-1}+\ldots+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}$

Show further that $y(x)=x e^{\mu x}$ is also a solution of $(1)$ if $\mu$ is a root of the polynomial $p(\lambda)$ of multiplicity at least 2 .

(b) By considering $v(t)=\frac{d^{2} u}{d t^{2}}$, or otherwise, find the general real solution for $u(t)$ satisfying

$\frac{d^{4} u}{d t^{4}}+2 \frac{d^{2} u}{d t^{2}}=4 t^{2}$

By using a substitution of the form $u(t)=y\left(t^{2}\right)$ in $(2)$, or otherwise, find the general real solution for $y(x)$, with $x$ positive, where

$4 x^{2} \frac{d^{4} y}{d x^{4}}+12 x \frac{d^{3} y}{d x^{3}}+(3+2 x) \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=x$