 Solution

Use the drift-removing substitution for a Dirichlet Sturm-Liouville problem, $z(x)=e^{-2x}u(x)$. It gives $u^{\prime \prime }+\lambda u=0$ with $u(0)=u(1)=0$. Positive $\lambda =k^2$ requires $\sin k=0$; zero or negative $\lambda$ yields only the trivial solution under both endpoint conditions. HenceMultiplication of the original equation by $e^{4x}$ gives $-(e^{4x}{z}^{\prime }{)}^{\prime }-4e^{4x}z=\lambda e^{4x}z$. Thus the appropriate weighted inner product has weight $e^{4x}$, andMultiplying each eigenfunction by $\sqrt{2}$ gives weighted norm one.