course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
41 |
|
3.II.15G |
2006 |
(a) Find the Fourier sine series of the function
$$f(x)=x$$
for $0 \leqslant x \leqslant 1$.
(b) The differential operator $L$ acting on $y$ is given by
$$L[y]=y^{\prime \prime}+y^{\prime}$$
Show that the eigenvalues $\lambda$ in the eigenvalue problem
$$L[y]=\lambda y, \quad y(0)=y(1)=0$$
are given by $\lambda=-n^{2} \pi^{2}-\frac{1}{4}, \quad n=1,2, \ldots$, and find the corresponding eigenfunctions $y_{n}(x)$.
By expressing the equation $L[y]=\lambda y$ in Sturm-Liouville form or otherwise, write down the orthogonality relation for the $y_{n}$. Assuming the completeness of the eigenfunctions and using the result of part (a), find, in the form of a series, a function $y(x)$ which satisfies
$$L[y]=x e^{-x / 2}$$
and $y(0)=y(1)=0$.