Paper 2, Section II, A

Differential Equations | Part IA, 2010

(a) Consider the differential equation

andnydxn+an1dn1ydxn1++a2d2ydx2+a1dydx+a0y=0a_{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 nNn \in \mathbb{N} and a0,,anRa_{0}, \ldots, a_{n} \in \mathbb{R}. Show that y(x)=eλxy(x)=e^{\lambda x} is a solution if and only if p(λ)=0p(\lambda)=0 where

p(λ)=anλn+an1λn1++a2λ2+a1λ+a0p(\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)=xeμxy(x)=x e^{\mu x} is also a solution of (1)(1) if μ\mu is a root of the polynomial p(λ)p(\lambda) of multiplicity at least 2 .

(b) By considering v(t)=d2udt2v(t)=\frac{d^{2} u}{d t^{2}}, or otherwise, find the general real solution for u(t)u(t) satisfying

d4udt4+2d2udt2=4t2\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(t2)u(t)=y\left(t^{2}\right) in (2)(2), or otherwise, find the general real solution for y(x)y(x), with xx positive, where

4x2d4ydx4+12xd3ydx3+(3+2x)d2ydx2+dydx=x4 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

Typos? Please submit corrections to this page on GitHub.