Differential equations with vectors

Question

I saw some equations like that $\operatorname{grad}(y(x,y,z)) + y(x,y,z) = 0.$ How can we solve it? What is this kind of differential equation called?

For example I found that:

$$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{V})=0. $$