 Solution

ID: past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2009/a/2/b/c/ii/solution

Here

sin^{2} x = (1 - cos 2 x) /2

belongs to the image. Matching the two coefficients using

L (A + B cos 2 x) = B π /2 + A π cos 2 x

gives

A = - 1/ (2 π)

B = 1/ π

. Consequently

y (x) = \frac{cos 2 x - 1/2}{π} + z (x), z ⊥ 1, z ⊥ cos 2 x .

There are infinitely many solutions, since any element of the infinite-dimensional kernel may be added.

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