Solving a constrained inhomogeneous first order ODE.

Question

Suppose that $A$ is a symmetric $n \times n$ matrix and $b \in \mathbf{R}^n$. I want to solve the following ODE, $$ \dot{x} + 2Ax +2b = 0, $$ with $x: \mathbf{R}_+ \to \mathbf{S}^{n-1}$, so the domain is the nonnegative reals and codomain is the the unit sphere. Is this possible with the implicit constraints on $x$? Are there good references that explain how to solve this type of ODE numerically and analyze it mathematically (i.e., stable equilibrium, etc.)?