 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2022/11/c/ii/solution

For

β^{2} = 1

, the reduced equation is

u^{''} + u = 1 + cos x

. The constant forcing has particular solution one, while the resonant cosine requires

u_{p} = (x /2) sin x

. Thus

y = e^{x} [A cos x + B sin x + 1 + \frac{x}{2} sin x], β = \pm 1.

The secular factor

x

is the signature of resonance. It must replace the divergent nonresonant coefficient

1/ (β^{2} - 1)

; substituting

β^{2} = 1

directly into that coefficient is invalid.

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