 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/ia/paper-2/7a/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 2 / 7A / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-07

Write the linear system of differential equations as

z^{'} = \frac{1}{t} A z + b, A = (4 - 1 - 2 5), b = (3 - 9) .

A homogeneous power

t^{λ} v

solves this Cauchy-Euler differential system precisely when

A v = λ v

. The eigenvalues are

3

and

6

, with eigenvectors

(2, 1)^{T}

and

(1, - 1)^{T}

. The homogeneous solution is therefore

C t^{3} (2, 1)^{T} + D t^{6} (1, - 1)^{T}

.

For a particular solution, put

z_{p} = t c

. Then

(I - A) c = b

, which gives

c = (3, 0)^{T}

. At

t = 1

, the two initial conditions become

2 C + D + 3 = 0

and

C - D = 0

, so

C = D = - 1

. Thus

x (t) = 3 t - 2 t^{3} - t^{6}, y (t) = t^{6} - t^{3}, t \geq 1 .

The two powers correspond to the two eigenvalues; the lower-degree particular term accounts for the constant forcing. Substitution gives both original right sides and the zero initial data.

 Read the full article

 New to topics? Read the docs here!