Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 8A / a

Solution

Write the system as

x^{'} = M x

, with

M = (2 - 1 5 - 2)

λ^{2} + 1

, so its eigenvalues are

i

and

- i

i

v = (5, - 2 + i)^{T}

. Thus

e^{i t} v

is a complex solution, and its real part and imaginary part give real solutions:

u (t) = (5 cos t - 2 cos t - sin t), w (t) = (5 sin t cos t - 2 sin t) .

Their initial vectors are

(5, - 2)^{T}

and

(0, 1)^{T}

5 \neq = 0

. Consequently they are linearly independent and span the two-dimensional solution space.

The general real solution is

(x (t) y (t)) = α (5 cos t - 2 cos t - sin t) + β (5 sin t cos t - 2 sin t) .

(2)

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