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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.

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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).

enter image description here

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 $

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    @RobertIsrael: Dear Sir. I am looking forward to hearing your interesting comments http://math.stackexchange.com/questions/175340/two-concepts-of-monotonicity2012-07-26
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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.