Eigenvalues and Eigenfunctions — Question 2

Question 2 – Solution

Strategy. Preserve the constant solution by checking the zero parameter before dividing by a frequency.

Step 1: Find the complete spectrum. Integration by parts for any eigenfunction gives $\lambda\int_0^L y^2dx=\int_0^L(y')^2dx\geq 0$, so negative eigenvalues are impossible. At zero, $y=A+Bx$ and the derivative conditions force $B=0$; nonzero constants are eigenfunctions. For $\lambda=k^2>0$, $y=A\cos(kx)+B\sin(kx)$; the left derivative condition forces $B=0$ and the right requires $\sin(kL)=0$. Thus $$\boxed{\lambda_n=(n\pi/L)^2,\quad E_n=\operatorname{span}\{\cos(n\pi x/L)\}, \quad n=0,1,2,\ldots.}$$

Step 2: Normalize the modes. The constant has squared integral $L$, while every positive-frequency cosine has squared integral $L/2$. The positive value at zero fixes the signs: $$\boxed{\phi_0=1/\sqrt L,\qquad \phi_n=\sqrt{2/L}\cos(n\pi x/L)\quad(n\geq 1).}$$

Step 3: Interpret the mean constraint. Integrating gives $\lambda\int_0^L y\,dx=-y'(L)+y'(0)=0$. Thus all modes with $\lambda\ne 0$ have zero mean. The only constant with zero mean is the zero function, so the additional constraint removes the zero eigenspace from the admissible nonzero functions; it preserves every higher mode.

Step 4: Test the two proposed functions. The function $x$ has derivative one at both ends, so it fails the boundary conditions. The function $g=1+\cos(\pi x/2)$ satisfies both derivative conditions, but $-g''=(\pi^2/4)\cos(\pi x/2)$ is not any constant multiple of $g$. Indeed, matching the constant term would require $\lambda=0$, which fails for the cosine term. A sum from different eigenspaces need not be an eigenfunction.