# 4 coupled ode

1. Oct 10, 2009

### Qyzren

Hi guys, i have 4-coupled ode's that are giving trouble

$$(1) \frac{dy_1}{dt}=y_2y_3-\mu y_1, \hspace{1cm} \\(2) \frac{dy_2}{dt}=y_1y_4-\mu y_2, \hspace{1cm} \\(3) \frac{dy_3}{dt}=1-y_1y_2, \hspace{1cm} \\(4) \frac{dy_4}{dt}=1-y_1y_2$$
I need to show that the steady state solutions are
$$y_1=\pm k, y_2=\pm k^{-1}, y_3=\mu k^2, y_4 = \mu k^{-1}$$ where $$\mu (k^2-k^{-2})=A$$ a const.
now in an earlier part of the question, I was able to show that $$y_3-y_4=A$$.
But trying to solve these coupled ODE's is giving trouble. I tried solving this in mathematica using DSolve as well, and mathematica doesn't seem to know how to do it either.

mathematica code:
$$\text{DSolve}\left[\left\{\text{y1}'[t]==-\mu \text{y1}[t]+\text{y2}[t] \text{y3}[t],\text{y2}'[t]==-\mu \text{y2}[t]+\text{y1}[t] \text{y4}[t],\text{y3}'[t]==1-\text{y1}[t] \text{y2}[t],\text{y4}'[t]==1-\text{y1}[t] \text{y2}[t]\right\},\{\text{y1}[t],\text{y2}[t],\text{y3}[t],\text{y4}[t]\},t\right]$$

PS: how do i get each of the differential equations on a new line? \\ and \newline didn't work.
each of the ODE's starting with the derivative should be on a new line

Last edited: Oct 10, 2009
2. Oct 10, 2009

### Dick

You don't have to solve the ODE's to find steady state solutions, do you? Just put all of the derivatives equal to zero. Now it's just an algebra problem.

3. Oct 10, 2009

### Qyzren

thanks, problem solved