= Solution
{c}
With $R=0$, substitute $h=X(x)T(t)$ into the dimensional <Boussinesq equation for an unconfined aquifer>:
$$
\phi XT'=KT^2(XX')',\qquad
\frac{\phi T'}{KT^2}=\frac{(XX')'}X=-\lambda.
$$
For a positive draining profile, take $\lambda>0$. Normalize $T(t^*)=1$, so $X$ has the dimensions of height. The temporal <ordinary differential equation> integrates to
$$
\boxed{T(t)=\frac1{1+(K\lambda/\phi)(t-t^*)}.}
$$
The spatial <ordinary differential equation> and <boundary conditions> are
$$
\boxed{(XX')'+\lambda X=0\quad(0<x<L),\qquad
X(0)=0,\quad X'(L)=0,\quad X>0\text{ on }(0,L].}
$$
Here $\lambda$ has units of inverse length with this normalization. The scale freedom between $X$ and $T$ is fixed by the chosen value $T(t^*)=1$; the amplitude of $X$, and hence $\lambda$, must be matched to the available water.
This <separable aquifer drawdown> is exact if the inherited height profile has the separated spatial shape. For a general rainfall-generated <initial condition>, it describes the late drought regime; it cannot reproduce an arbitrary profile at the instant rainfall stops merely by choosing one temporal constant.
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