OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 13A / b / Solution

Because the parameter enters through

λ w y

,

L (y; λ_{0}) = L (y; λ) - (λ - λ_{0}) w y = y^{m + 1} - ε^{m} μ w y .

Use this as

f

in part (a). The necessary orthogonality condition is

0 = \int_{0}^{1} y_{0} (y^{m + 1} - ε^{m} μ w y) d x .

With

y = ε y_{0} + ε^{2} y_{1}

and

μ = O (1)

,

y^{m + 1} = ε^{m + 1} y_{0}^{m + 1} + O (ε^{m + 2}), w y = εw y_{0} + O (ε^{2}) .

Consequently

0 = ε^{m + 1} [\int_{0}^{1} y_{0}^{m + 2} d x - μ \int_{0}^{1} w y_{0}^{2} d x] + O (ε^{m + 2}) .

The normalization makes the second integral one. Divide by

ε^{m + 1}

to obtain

μ = \int_{0}^{1} y_{0}^{m + 2} d x + O (ε) .

This is the leading nonlinear eigenvalue shift in a Sturm-Liouville problem.

Solved by gpt-5.6-sol high.

 Ancestors (11)