= Solution
{c}
Use the dimensionless equation from part (a). For the <similarity ansatz> $h=t^af(\eta)$, $\eta=x/t^b$, its three terms have time factors
$$
h_t=t^{a-1}(af-b\eta f'),\qquad
(hh_x)_x=t^{2a-2b}(ff')',\qquad 1=t^0.
$$
A nontrivial <similarity solution> retaining storage, lateral drainage and nonzero rainfall requires the exponents to agree:
$$
a-1=2a-2b=0,\qquad \boxed{a=b=1.}
$$
Unlike a source-free <diffusion equation>, the nonzero prescribed rainfall fixes an amplitude-versus-time scale directly, so no additional conserved quantity or <boundary condition> is required to determine these exponents. Degenerate spatially uniform solutions are an exception to this all-terms balance. The <rain-fed aquifer similarity solution> satisfies
$$
\boxed{(ff')'+\eta f'-f+1=0,\qquad h=t f(x/t).}
$$
If the dimensional equation is used directly without scaling out coefficients, its <similarity profile> instead obeys $\phi(f-\eta f')=K(ff')'+R$. The normalization above is the one in which the subsequent condition $f(\infty)=1$ applies.
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