= Solution
Try $\rho=t^af(\eta)$, $\eta=x/t^b$. The boundary data force $a=3$ and $f(0)=1$. The time <derivative> scales as $t^{a-1}$, advection as $t^{2a-b}$ and the source as $t^{a+c}$. Balancing all three gives $c=-1$ and $b=a+1=4$. Substitution then yields
$$
3f-4\eta f'+ff'=f,\qquad
\boxed{c=-1,\quad\eta=x/t^4,\quad(f-4\eta)f'+2f=0,\quad f(0)=1.}
$$
For the pointwise initial trace at any fixed $x>0$, the far-profile requirement is $f(\eta)=o(\eta^{3/4})$ as $\eta\to\infty$ on any smooth branch extending that far; a compact-support continuation also meets it. Solving or selecting a <shock> continuation was not required. The time-dependent source is singular at $t=0$, so the similarity reduction is stated on $t>0$.
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