# A Difficult Method of Characteristics Problem

1. Jul 4, 2008

### Hoplite

I'm trying to find characteristic curves for the following ordinary differential equations:

$$\frac{d\kappa }{dt} = \mu \kappa \xi (1-\chi ), \qquad && \frac{d\chi }{dt} = \mu \chi \xi (\chi -1), \qquad \frac{d\zeta }{dt} = \lambda \zeta (1-\zeta ) + \mu \zeta ( 1-\xi ), \qquad && \frac{d\xi }{dt}= \lambda \xi (\zeta -1) +\mu \xi (\xi -1).$$

My purpose is to use them to solve the following:

$$\frac{\partial \mathcal{P}}{\partial t} + \mu \kappa \xi (1-\chi )\frac{\partial \mathcal{P}}{\partial \kappa } + \mu \chi \xi (\chi -1) \frac{\partial \mathcal{P}}{\partial \chi } + [ \lambda \zeta (1-\zeta ) + \mu \zeta ( 1-\xi )] \frac{\partial \mathcal{P}}{\partial \zeta } \nonumber + [\lambda \xi (\zeta -1) +\mu \xi (\xi -1)]\frac{\partial \mathcal{P}}{\partial \xi } = \lambda \zeta (\kappa -1 )\mathcal{P}$$

I can see that the first two ODEs above together give $$\kappa \chi = \mbox{constant}$$ and $$\zeta \xi = \mbox{constant}$$, but I figure that I'll need to derive 4 constants for use in the method of characteristics.

If anyone can shed some light on this, I'll be quite impressed.