Probelm 7.60 of Griffiths' Introduction to Electrodynamics, 4th ed, says:
Suppose $\mathbf J(\mathbf r)$ is constant in time but $\rho(\mathbf r, t)$ is not—conditions that might prevail, for instance, during the charging of a capacitor.
(a) Show that the charge density at any particular point is a linear function of time: $$ \rho(\mathbf r, t) = \rho(\mathbf r, 0) + \dot \rho(\mathbf r, t)t, $$ where $\dot \rho(\mathbf r, 0)$ is the time derivative of ρ at t = 0.
(b) Show that $$ \mathbf B(\mathbf r) = \frac{\mu_0}{4\pi} \int \frac{\mathbf J(\mathbf r')\times \hat{\boldsymbol{\ell}}}{\ell^2} d \tau' $$ obeys Ampère’s law with Maxwell’s displacement current term.
Where $ \ell = |\boldsymbol{\ell}|, \quad \boldsymbol{\ell} = \mathbf {r - r'}, \quad \boldsymbol{\hat \ell} = \boldsymbol{\ell}/\ell, \quad $ (since I don't know how to type his cursive $r$ in LateX.)
So the (a) is solved by invoking (as hinted) the continuity equation. But for (b) I have a different answer, namely $$ \mathbf B(\mathbf r) = \frac{\mu_0}{4\pi} \int \frac{(\mathbf J(\mathbf r') + \varepsilon_0 \dot{\mathbf E}(\mathbf r',0) )\times \hat{\boldsymbol{\ell}}}{\ell^2} d \tau' $$ and this is my justification:
Since the charge density is as above and $\mathbf E(\mathbf r,t)$ is given (as in the manual) by $$ \mathbf E(\mathbf r,t) = \frac{1}{4\pi\varepsilon_0} \int \frac{\rho(\mathbf r',t) \hat{\boldsymbol{\ell}}}{\ell^2} d\tau' = \mathbf E(\mathbf r,0) + \dot{\mathbf E}(\mathbf r,0)t $$ then the two Maxwell's equations \begin{align*} \nabla \cdot \mathbf B &= 0 & \nabla \times \mathbf B = \mu_0 \mathbf J + \mu_0\varepsilon_0 \partial_t \mathbf E, \end{align*} with $\mathbf J$ and $\partial_t \mathbf E$ functions only of position (but not of time) and with $\nabla\times \mathbf E = 0$ (by direct computation), are decoupled from the other two and we have the solution for $\mathbf B$ as the Bio-Savart Law with $\mathbf J(\mathbf r')$ replaced with $\mathbf J(\mathbf r') + \varepsilon_0 \dot{\mathbf E}(\mathbf r')$.
Otherwise where is my mistake ?