Solve system of nonlinear differential equations 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.
 A: Solutions are likely to hit a singularity when the matrix $M$ with rows $(a_i + P_i x/\|P_i x\|)^T$ becomes singular or when $x$ approaches the origin.  In your $n=2$ example with $u = x/\|x\|$, I get $\det(M) = 1 + \sum_j u_j$, and you'll get a singularity if that hits $0$.  That does happen, e.g. with initial conditions $x_1(0)=1$, $x_2(0)=0$, at approximately $t= 1.2464504$ according to Maple's dsolve(..., numeric). 

EDIT: Hmm, in fact $x_1 - x_2$ is constant in this system, and you get a singularity when $x_1 = 0$, $x_2 < 0$.  If $d = x_1 - x_2$, the system has closed-form implicit solutions
$$ t+\ln  \left( \left( 2 x_1 \left( t \right) +d
 \right) \sqrt {2}/2+\sqrt {2\, \left( x_1 \left( t \right) 
 \right) ^{2}+2\,x_1 \left( t \right) d+{d}^{2}}/2 \right) \sqrt {2
}/2 +\ln  \left( 2\, \left( x_1 \left( t \right)  \right) ^{2}+
2\,x_1 \left( t \right) d+{d}^{2} \right)/2 +c=0
$$
A: If you kow FORTRAN go to netlib.org and get numerical package for nonlinear systems of equations. Usually you need a good approximation to the solution to start the search. 
