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A Difficult Method of Characteristics Problem

  1. Jul 4, 2008 #1
    I'm trying to find characteristic curves for the following ordinary differential equations:

    [tex]\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). [/tex]

    My purpose is to use them to solve the following:

    [tex]\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} [/tex]


    I can see that the first two ODEs above together give [tex] \kappa \chi = \mbox{constant} [/tex] and [tex] \zeta \xi = \mbox{constant} [/tex], 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.
     
  2. jcsd
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