Hi I'm trying to compare my leading order estimate for eigenvalues $\lambda_n$ for the following problem. $$ \frac{d^2y(x)}{dx^2} + \frac{\lambda^2}{x^2}y(x) = 0\;\text{ with } y(1) = 0\text{ and } y(e) = 0.\label{1}\tag{1}$$
From the WKB method I got the result $\lambda_n = n\pi$ with eigenfunction $$ y_n(x) = B\sqrt{x} \sin(n\pi \log(x)). $$
Then I solved \eqref{1} analytically by letting $y=x^m$ and then, proceeding in the usual way, applying boundary conditions I get the result $$ y_n(x) = B\sqrt{x} \sin (\frac{\sqrt{1-4\lambda^2}}{2} \log(x)). $$ Then, if I apply $y(e) = 0,$ for non-trivial solutions I obtain $$ \lambda_n = \sqrt{\frac{1}{4} - n^2\pi^2}. $$
Have I made an error here? I am trying to compare the leading order, but the analytical solution appears to give imaginary eigenvalues. Hopefully someone can steer me to the correct solution or fix my error.
Best.
Your analytical solution missed a sign change in computing the complex roots. It should be $$ y=\sqrt{x}\sin\left(\frac{\sqrt{4λ^2-1}}2\ln(x)\right). $$ In detail, the characteristic equation for $m$ is $$0=m(m-1)+λ^2=(m-\frac12)^2+λ^2-\frac14.$$ For $λ>\frac12$ the solutions are a complex conjugate pair $$m=\frac12\pm i\sqrt{λ^2-\frac14}.$$