Techniques of applied maths: Eigenfunction expansions I

Eigenvalues and eigenfunctions of a BVP

Consider the eigenvalue problem on $0 \leq x \leq 1$, \begin{gather} y'' + 2y' + (1 + \lambda) y = 0, \\ y'(0) + y(0) = 0, \\ y'(1) + y(1) = 0. \end{gather}

a) Eigenvalues and eigenfunctions of the BVP

Assuming $\lambda$ to be a positive constant, what is the general solution of the homogeneous ODE. Apply the boundary conditions to determine the eigenvalues and eigenfunctions.


b) Eigenvalues and eigenfunctions of the adjoint problem

What is the adjoint problem? Obtain the adjoint eigenfunctions


c) Placing the problem into SL form

What are the functions $p$, $q$, $r$, that put this problem in standard Sturm-Liouville form: \[ \left( p(x) \frac{dy}{dx}\right)' + q(x) y = -\lambda r(x) y \]

Verify that the expected orthogonality conditions (i.e. the one in terms of $y$'s and $r$ and the one in terms of $y$'s and $w$'s) are satisfied by direct integration with the eigenfunctions.


d) Eigenfunction expansion

Use the eigenfunctions and their adjoints to obtain the coefficients in the eigenfunction expansion \[ y(x) = \sum_{k = 0}^\infty c_k y_k \]

for the solution of the problem \begin{gather*} y'' + 2y' + 2y = 1 \\ y'(0) + y(0) = 2 \\ y'(1) + y(1) = 3. \end{gather*}