## System of differential equations with variable coefficients [closed]

What can be done with this?

$\frac{dx}{dt} = \alpha(x+y)(M-x)$

$\frac{dy}{dt} = \alpha(N-y)x$

$x(0) = c$, $y(0) = d$

$\alpha, M, N, c, d$ are positive reals. The system dynamics exists only until $x$ reaches $M$ or $y$ reaches $N$.

Can we get $x$ and $y$ as functions of $t$?

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I'm voting to close. The fundamental reason is that you have not even done the minimum effort to present this in a way that is remotely natural. For example, $\alpha$ is inessential, since you could rescale $t$. It gives the impression that you are asking people to do your work for you and that's simply not friendly. – JosÃ© Figueroa-O'Farrill Aug 13 2010 at 1:23

## closed as too localized by JosÃ© Figueroa-O'Farrill, S. Carnahan♦Aug 13 2010 at 14:16

I have no idea where this system comes from or what you have tried or are interested in trying.

My initial thought is to look at this numerically using a simple finite-difference approximation for the time derivative in order to express the functions $x$ and $y$ in terms of $t$.

A central-difference approximation for the time derivatives looks like this: $$\frac{dx}{dt} \approx \frac{x_{i+1} - x_{i}}{\Delta t}, \text{ and similarly for }\frac{dy}{dt}$$

where $\Delta t = t_{i+1} - t_{i}$ is a time step we can choose later, and $x_{i+1}$ is a symbol for $x(t_{i+1})$.

So, the system can be expressed as:

\begin{align} x_{i+1} &= \Delta t \left[ \alpha(x_{i} + y_{i})(M + x_{i})\right] + x_{i} \\ & \\ y_{i+1} &= \Delta t \left[ \alpha(N - y_{i})x_{i}\right] + y_{i} \end{align}

This RHS of this formulation is expressed in terms of known quantities (coming from the initial conditions). We also essentially have expressions for $x(t)$ and $y(t)$, but are not in general reliable in the long term.

If this looks interesting to you, take this question over to math.stackexchange.com and see what they (we) come up with. We can run this through matlab or whatever if you like. I am not so sure it is appropriate here.

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 Thanks for your response. I was hoping that this set of equations might be in some form for which a closed form solution exists and which is well-recognised by those in the field. I have looked around, but not seen one. Thanks for your suggestion all the same. – MAKCL Aug 13 2010 at 2:20

You have two (nonnegative) fixed points (0,0) and (M,N). The first one is a saddle and the second one is a stable node. The (nonnegative part of the) unstable manifold of the fisrt one converges to the second one. A trivial dynamical analysis of the phase plane can be done easily. I recommend the plotting of the solutions in the phase plane using Mathematica's NDSolve.

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