 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/15/a/i/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 15 / a / i / Solution by

Codex  0 2026-10-18

For the homogeneous linear differential equation, the characteristic equation is

r^{2} - 2 r + 2 = 0

, with polynomial roots

1 \pm i

. Its homogeneous solution is therefore

e^{x} (A cos x + B sin x)

.

Use the method of undetermined coefficients for a particular solution

y_{p} = a x^{2} + b x + c

. Substitution gives

2 a x^{2} + (2 b - 4 a) x + (2 a - 2 b + 2 c) = 2 x^{2} .

Equating coefficients yields

a = 1

,

b = 2

and

c = 1

. Hence the general solution is

y = e^{x} (A cos x + B sin x) + x^{2} + 2 x + 1.

 Read the full article

 New to topics? Read the docs here!