= Solution
Substitution into the <diffusive flux> gives
$$
J(x,t)=-e^{-\lambda t}D(x)\bigl(u'(x)+\beta V'(x)u(x)\bigr).
$$
The <continuity equation> for this <drift-diffusion equation with a potential> therefore reduces, after cancelling the nonzero exponential, to
$$
\boxed{\bigl[D(u'+\beta V'u)\bigr]'=-\lambda u.}
$$
The reflecting <boundary conditions> become $D(u'+\beta V'u)=0$ at both endpoints. Thus each solution of this <eigenvalue problem> supplies a separated solution of the original <diffusion equation>.
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