Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 1 / b / iii / Solution

Expand the solution and the forcing in the orthonormal Sturm-Liouville eigenfunction expansion. The forcing coefficients are

b_{n} = ⟨ y_{n}, 1 ⟩_{w} = 2 \int_{0}^{1} sin (nπ t) d t = \frac{2 [ 1 - ( - 1 ) ^{n} ]}{nπ} .

Every

λ_{n}

is nonzero, so the solution coefficient is

b_{n} / λ_{n}

. Thus

y (x) = n = 1 \sum \infty \frac{2 [ 1 - ( - 1 ) ^{n} ]}{nπ ( n ^{2} π ^{2} - 1 )} sin (nπ lo g x) = n \geq 1 n odd \sum \frac{4 sin ( nπ lo g x )}{nπ ( n ^{2} π ^{2} - 1 )} .

As an independent check, solving

Y^{''} + Y = - 1

directly with the two endpoint conditions gives

y (x) = \frac{c o s ( l o g x - \frac{1}{2} )}{c o s ( \frac{1}{2} )} - 1.

It satisfies both Dirichlet boundary conditions and the forced ordinary differential equation.

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