= Solution
Project the <initial condition> onto the orthogonal spatial <eigenfunctions>. Their squared norms in the ordinary <inner product> are
$$
\int_0^1e^{2x}\,dx=\frac{e^2-1}{2},\qquad
\int_0^1V_n(x)^2\,dx=\frac{n^2\pi^2+1}{2}.
$$
The latter follows from the integrals of $\cos^2(n\pi x)$ and $\sin^2(n\pi x)$ being $1/2$, with the sine-cosine cross term zero. Therefore
$$
\boxed{c_0=\frac2{e^2-1}\int_0^1f(x)e^x\,dx,\qquad
c_n=\frac2{n^2\pi^2+1}\int_0^1f(x)\left[n\pi\cos(n\pi x)+\sin(n\pi x)\right]dx.}
$$
For square-integrable data the expansion converges to the data in the <L2 norm> at $t=0$; for $t>0$ the decaying high modes are smoothed. Initial data with zero $e^x$ projection have no growing term.
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