I am trying to solve a large system of differential equations. Ideally, I would like to solve it exactly, but if not, can anyone suggest me a numerical method?
In all its generality, the system I am trying to solve is like this: (here, $x = x(t) \in R^n$, and $\dot x = dx/dt$)
$$ (a_i + P_ix/\Vert P_ix \Vert)^T \dot x = -\Vert P_ix \Vert $$
for $i = 1,\ldots,n$. Here all $P_i$ are positive definite matrices, and the set of $a_i$ is linearly independent. Also, $\Vert . \Vert$ is the 2-norm.
It would help me a great deal if someone can help me to solve even a highly restricted special case of it, where $n=2$, $a_i = e_i$ (the $i$-th vector of the canonical basis), and $P_i = I$ for all $i$. Namely, this system:
$$ (e_i + x/\Vert x \Vert)^T \dot x = -\Vert x \Vert $$ for all $i$.
Thanks a lot, Daniel.