In a x,y,z cartesian coordinate system with the z-axis positive downwards, equations (1) and (2) are "equivalent", in that they both describe how the vector field $\mathbf{v}$ is governed by the vector $(\nabla p - \rho \mathbf{g})$: $$\tag{1} \mu \nabla ^2 \mathbf{v} = \nabla p - \rho \mathbf{g}; \ \ \nabla \cdot \mathbf{v}=0$$ $$\tag{2} \mathbf{v}=-\frac{f(\mathbf{x})}{\mu}(\nabla p - \rho \mathbf{g}); \ \ \nabla \cdot \mathbf{v}=0$$
However, after distributing the leading negative sign in Eqn(2), we have $(-\nabla p + \rho \mathbf{g})$. Thus, the $\nabla p$ and $\rho \mathbf{g}$ terms have opposite signs in Eqns(1) and (2). This leads me to think that Eqns(1) and (2) are not "equivalent". I think my confusion is due to Eqn(1) is written for $\nabla ^2 \mathbf{v}$ rather than for $\mathbf{v}$ as is for Eqn(2). Does the quantity $\nabla ^2 \mathbf{v}$ in Eqn(1) lead to a negative quantity, where the negative sign can be distributed to the vector $(\nabla p - \rho \mathbf{g})$, and if 'yes' can you show me how?