 Solution

ID: past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2011/a/2/b/ii/solution

Since

λ_{1} = - 4/5

L + \frac{4}{5} I

is the span of

ψ (x) = cos (3 x /2)

⟨ f, ψ ⟩ = 0

for solvability. If it holds, an arbitrary multiple of

ψ

can be added, so a solution is not unique.

For

f = cos x

\int_{0}^{π} cos x cos (3 x /2) d x = \frac{6}{5} \neq = 0,

so case (A) has no solution. For

f = sin x sin (3 x /2)

\int_{0}^{π} f (x) ψ (x) d x = \frac{1}{2} \int_{0}^{π} sin x sin 3 x d x = 0

by sine-function orthogonality. Thus case (B) has solutions, and they are nonunique.

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