= Solution
Use the <drift-removing substitution for a Dirichlet Sturm-Liouville problem>, $z(x)=e^{-2x}u(x)$. It gives $u''+\lambda u=0$ 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. Hence
$$
\boxed{\lambda_n=n^2\pi^2,\qquad z_n(x)=e^{-2x}\sin(n\pi x),\qquad n=1,2,\ldots.}
$$
Multiplication of the original equation by $e^{4x}$ gives $-(e^{4x}z')'-4e^{4x}z=\lambda e^{4x}z$. Thus the appropriate <weighted inner product> has weight $e^{4x}$, and
$$
\boxed{\int_0^1e^{4x}z_m(x)z_n(x)dx=\int_0^1\sin(m\pi x)\sin(n\pi x)dx=\frac12\delta_{mn}.}
$$
Multiplying each eigenfunction by $\sqrt2$ gives weighted norm one.
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