 Solution

ID: past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2008/a/2/e/solution

The integral equation says precisely

h = L^{- 1} z

with the homogeneous mixed endpoint conditions. Since the supplied

h

satisfies those conditions, apply

L

directly:

z (x) = L h = h^{''} + h = x^{2} - 2 x + 2.

No integration is required. The absence of a zero eigenvalue established in (b)–(c) also proves uniqueness of this recovered forcing.

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