# Is there an analytical solution to this system of ODEs?

I have been working on a problem, and reached a part where I have a system of ODEs. Let $x(t) \in \mathbb R^n$ be a univariate function, and $\dot x(t)$ be its derivative. I will drop the $(t)$ for ease of notation.

Let $m < n$, and let $A$ be a full rank $m \times n$ matrix. Also, for $i = 1,\ldots,n-m$ let $L_i$ be a subset of $\{1,2,\ldots,n\}$ and $1 \le k_i \le n$, with $k_i \not\in L_i$. Let $x^{L_i} \in \mathbb R^{|L_i|}$ be the components of $x$ corresponding to the indices in $L_i$. The same for $\dot x$. The system of ODEs I am trying to solve has the following form:

\begin{align*} A\dot x &= 0 & \\ 2t \langle x^{L_i}, \dot x^{L_i}\rangle - \dot x^{k_i} &= -\Vert x^{L_i} \Vert && \mbox{$\forall i = 1,\ldots,n-m$} \end{align*}

I don`t know if my notation is confusing. If it is, please let me know. I am almost sure that I won't be able to solve it analytically, but a man can dream :)

Given that an explicit solution is rather unlikely (you have irrational nonlinearity in $\|\cdot \|$), what is the purpose of your investigation of this ODE system? Are you interested in asymptotic long-term behavior, stability with respect to initial values, or in approximate solutions? – user53153 Jan 28 at 3:54
Both stability with respect to initial values and approximate solutions would be very interesting. Specially approximate solutions! In my application I will only need values of $x(t)$ for $t \in [0,1]$, so asymptotic behavior is not really important. – Daniel Fleischman Jan 28 at 10:32
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