# convergence of ODE [closed]

I have 2 coupled linear ODEs. I used Mathematica to solve for analytical solution. But the analytical solution looks too complicated. I only need to derive some monotonicity property of the solution. I don't know if it is doable without looking at the explicit expression of the solution.

\begin{equation} \begin{cases} &\dot{y}=(c_1-c_2c_3)(1-B)/B-(1-B)[c_2(1-c_3)-(y-x)-(c_1-c_2c_3)(1-B)/B]\\ &\dot{x}=-x+(1-B)[c_2(1-c_3)-(y-x)-(c_1-c_2c_3)(1-B)/B] \end{cases} \end{equation} The initial condition is that $x(0)>0$, $y(0)>0$ and $x(0)+y(0)<1$.

Thoses $c_i$'s are constants such that $c_1\in(0,0.5)$, $c_2\in(0,1)$, $c_3\in(0.5,1)$. And $c_2c_3<c_1<c_2$. $B$ is also a constants such that $B\in(0.5,1-c_1)$.

I can set $\dot{x}=0=\dot{y}$ and then solve the linear system of equations to get the fixed point of this ODE, which is unique. But how can I show that $x(t)$ and $y(t)$ are monotonically converge to that fixed point?

• Phase plane analysis. – Robert Israel Apr 27 '16 at 17:52
• This is a simple 2D linear ODE with constant coefficients and is better suited at math.stackexchange.com – Jaap Eldering Apr 27 '16 at 19:40