Question

Find the eigenvalues and eigenfunctions of the given Sturm-Liouville lems: a. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\) b. \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\).

Short answer

The eigenvalues for system a are \(\lambda_n=n^2\) and for system b are \(\lambda_m=m^2\). The eigenfunctions for system a are \(y_n = B \sin(nx)\) and for system b are found by substituting the eigenvalues and using the boundary conditions.

Step-by-step solution

Step 1a: Identify the boundary conditions for system a

The boundary conditions are given as \(y^{\prime}(0)=0\) and \(y^{\prime}(\pi)=0\). These will be important to find the eigenvalues.

Step 2a: Determine the eigenvalues for system a

To solve the differential equation, \(y^{\prime \prime}+\lambda y=0\), we assume a trial solution of \(y=Ae^{i\sqrt{\lambda}x } + B e^{-i\sqrt{\lambda}x}\). Applying the boundary conditions will yield the values for A and B, and hence give the eigenvalues. After arranging, the characteristic equation becomes \(\lambda n^2\), so the eigenvalues are \(\lambda_n=n^2\), where \(n=0,1,2,...\).

Step 3a: Determine the eigenfunctions for system a

Having the eigenvalues, the eigenfunctions can be found for each \(n\) by substituting the values of \(n\) into \(y_n\). This yields the eigenfunctions, \(y_n = A \cos(nx) + B \sin(nx)\). With the boundary conditions from step 1, the final eigenfunctions are \(y_n = B \sin(nx)\).

Step 1b: Identify the boundary conditions for system b

The boundary conditions are given as \(y(1)=0\) and \(y\left(e^{2}\right)=0\). These will be important to find the eigenvalues.

Step 2b: Determine the eigenvalues for system b

To solve the differential equation, \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0\), we assume a trial solution similar to system a, and apply the boundary conditions. After doing so, the characteristic equation becomes \(\lambda m^2\), where \(m\) is any integer. So the eigenvalues are \(\lambda_m=m^2\), where \(m=1,2,3,...\).

Step 3b: Determine the eigenfunctions for system b

The eigenfunctions can then be found for each \(m\) by substituting into \(y_m\). This gives the general estimated form of the eigenfunctions. Applying the boundary conditions from step 1 gives the eigenfunctions of system b.