# Solving a constrained inhomogeneous first order ODE.

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.)?

• You cannot solve the ODE with such constraints. However, you can write the DE satisfied by the directions of the solutions to the original DE. If my calculations are correct, that should be: $$\frac{d}{dt}\frac{x(t)}{\lVert x(t) \rVert} = -2 \left( (\langle x(t), A x(t) \rangle + \langle x(t), b \rangle ) \frac{x(t)}{\lVert x(t) \rVert^2} - A x(t) - b \right).$$ – user539887 May 3 '18 at 8:12
• And if $b = 0$ the equilibria of the "new" DE are just eigenvectors of the matrix $A$. Certainly there are references, and, hopefully, some expert on control theory will read your post and give them. I can (and do) upvote your post only. – user539887 May 3 '18 at 8:19
• @user539887, yes I hope so, I am very interested in solution methods and how to understand equilibria. – Drew Brady May 3 '18 at 21:41