 Solution

ID: past-exam-of-the-differential-equations-2-course-of-the-university-of-oxford/2023/1/c/ii/solution

The nonzero endpoint data contribute to the Fredholm alternative condition. With

w = e^{x} sin x

, the same integration by parts identity gives

\int_{0}^{π} w L y d x = [w y^{'} - w^{'} y + 2 w y]_{0}^{π} = - w^{'} (π) y (π) + w^{'} (0) y (0) = 1.

For the specified forcing the left side is

A \int_{0}^{π} sin^{2} x d x = A π /2

. Hence

A = \frac{2}{π} .

To verify sufficiency, set

y = e^{- x} u

. Direct differentiation gives

L y = e^{- x} (u^{''} + u)

. At this value of

A

u^{''} + u = (2/ π) sin x

- (x / π) cos x

. The complete family obeying the two boundary conditions is

y (x) = e^{- x} [(1 - \frac{x}{π}) cos x + C sin x], C \in R .

The arbitrary constant records the one-dimensional kernel; existence does not imply uniqueness here.

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