# Help for analytically solving a simple-looking nonlinear ODE system

I sincerely wish somebody can help me to analytically solve following nonlinear ODE system. $x(t)$ $y(t)$ $z(t)$ are 3 functions of $t$, and I will ignore to write $t$ in the system. It's derived from multi-compartment model, with $x^2$ $y^2$ and $z^2$ more...

$$\left\{ \begin{array}{l} x\prime=-x^2+x+y+z\\ y\prime=-y^2+x+y+z\\ z\prime=-z^2+x+y+z \end{array} \right.$$

Furthermore, this system can be extended with with $n$ functions...

I do need close-forms of $x(t)$ $y(t)$ $z(t)$ so that I can do my later tasks... I will appreciate you a lot if you can offer me any solutions, hopefully a general solution for any $n$.

btw, I tried laplace transform, but after transmission, the system is still too hard to solve...

Thanks a lot! : )

## 1 Answer

Due to the symmetry of the system there is a one-parametric family of solutions: $$x(t)=y(t)=z(t)=\frac{3 e^{3 t}}{e^{3 c_1}+e^{3 t}}.$$ For $n$ equations $$x_1(t)\ldots=x_n(t)= \frac{n e^{n t}}{e^{n c_1}+e^{n t}}.$$

• Dear, thank you so much for the instant reply!!!that's really a nice solution! I appreciate a lot, but sorry for bothering, actually $x(t) y(t) z(t)$ are with coefficients. E.g., $$\left\{ \begin{array}{l} x\prime=-A\cdot x^2+B\cdot x+C\cdot y+D\cdot z\\ y\prime=-E\cdot y^2+F\cdot x+G\cdot y+H\cdot z\\ z\prime=-I\cdot z^2+J\cdot x+K\cdot y+L\cdot z \end{array} \right.$$, so how should the solution be then? Thanks a lot! : ) – Dobby Oct 9 '11 at 15:05