# Solving differential equation with vector magnitude.

I am trying to solve a system of three coupled differential equations. I managed to simplify them using a matrix. $$\newcommand{\myMatrix}[1]{\bm{\mathit{#1}}} \frac{d\vec{x}}{dt}=\pmb{A}\left| \vec{x} \right|\vec{x}-\vec{d}$$ Where $$\pmb{A}$$ is a constant $$3\times3$$ matrix and $$\vec{d}$$ is a constant vector. I know there are ways to solve this if it weren't for the vector magnitude. Does anybody have any idea how to solve it with the vector magnitude?

• This seems impossible. Even in the 2D case with $A$ a constant, that is, a ballistic shot with air friction, you only get one first integral. – LutzL Aug 29 at 20:12
• @LutzL Out of curiosity how would you solve the first integral in the 2D case? – Jonas Aug 30 at 7:38
• This is a wunderkind topic (actually, there were previous published results in the 1970s, I imagine that this calculation was done repeatedly earlier, there is no big trick to it). See math.stackexchange.com/q/150242/115115 – LutzL Aug 30 at 7:58