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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?

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    $\begingroup$ 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. $\endgroup$ – LutzL Aug 29 at 20:12
  • $\begingroup$ @LutzL Out of curiosity how would you solve the first integral in the 2D case? $\endgroup$ – Jonas Aug 30 at 7:38
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    $\begingroup$ 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 $\endgroup$ – LutzL Aug 30 at 7:58

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