Past exam of the differential equations and applications course of the University of Oxford / 2009 / A / 2 / B / b

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- Solution b

Solution

Write

c (x) = cos 2 x

. The separated kernel gives

L y = c (x) \int_{0}^{π} y (t) d t + \int_{0}^{π} c (t) y (t) d t .

Thus its image is

E = span {1, c}

. Since

L 1 = π c

and

L c = π /2

, its matrix on this basis is

(0 π π /2 0)

. Therefore

λ_{\pm} = \pm \frac{π}{2}, y_{\pm} (x) = 1 \pm 2 cos 2 x .

Both eigenspaces have dimension one, and there are no further nonzero eigenvalues because every such eigenfunction belongs to the image. The zero eigenspace is

E^{⊥}

, which is infinite-dimensional. One explicit nonzero example is

y (x) = sin 2 x

, orthogonal to both

1

and

cos 2 x

.

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