 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017/7/b/solution

Im [(4 x + 2 i) e^{i x}] = 4 x sin x + 2 cos x

, a suitable complex forcing is

f (x) = (4 x + 2 i) e^{i x}

. Solve

u^{''} + 2 u = f

u = p (x) e^{i x}

reduces the equation to

p^{''} + 2 i p^{'} + p = 4 x + 2 i .

p = a x + b

gives

a = 4

and

b = - 6 i

. Thus

y_{p} (x) = Im [(4 x - 6 i) e^{i x}] = 4 x sin x - 6 cos x .

For a direct check,

(4 x sin x)^{''} + 2 (4 x sin x) = 8 cos x + 4 x sin x

, while

(- 6 cos x)^{''} + 2 (- 6 cos x) = - 6 cos x

. Their sum is the required forcing. Adding

A cos (2 x) + B sin (2 x)

gives the full real solution if needed.

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