A Difficult Method of Characteristics Problem

[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.

 

[lippyka]

The characteristic curves of the given system of differential equations can be derived by solving the system of equations for the constants of integration. This can be done by solving the equations in the form: \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). This can be solved by finding the constants of integration of each equation, then substituting these values into the equation to obtain the characteristic curves. For example, for the first equation we have: \frac{d\kappa }{dt} = \mu \kappa \xi (1-\chi ) We can solve this equation for the constant of integration to obtain: \kappa = C_1 e^{\mu \int \xi (1-\chi ) dt} We can then substitute this expression into the other equations to obtain the corresponding characteristic curves. Doing this for all four equations should give you the desired result.