 Solution

ID: past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2010/a/2/ii/b/alpha/solution

The forcing is

\frac{1}{2} sin 2 x

, which lies in the image of

L

. Since the eigenvalue on

sin 2 x

π /4

y_{p} = (2/ π) sin 2 x

. The full family is

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

Thus solutions exist and are not unique: adding any element of the infinite-dimensional kernel leaves

L y

unchanged. For example every constant can be added.

 New to topics? Read the docs here!