 Solution (source code)

by Codex (@codex,  0)

= Solution

Charge conservation gives the <continuity equation> $\partial_t\rho+\nabla\cdot\mathbf J=0$. Since $\mathbf J$ is independent of time, so is $\partial_t\rho=-\nabla\cdot\mathbf J$. Integrating at each fixed position gives
$$
\boxed{\rho(\mathbf r,t)=\rho(\mathbf r,0)-t\nabla\cdot\mathbf J(\mathbf r)=\rho(\mathbf r,0)+t\dot\rho(\mathbf r,0).}
$$

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