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I'm trying to solve the following PDE using the method of characteristics.

$$\frac{\partial F}{\partial t}+(a(x-1)+bx(x-1)+cx(y-1))\frac{\partial F}{\partial x}+(a(y-1)+by(x-1)+cy(y-1))\frac{\partial F}{\partial y}=(b(x-1)+c(y-1))NF,$$

where $a, b, c, N$ are nonnegative constants. (If this is too difficult, we can assume $\partial F/\partial t=0$ and get a steady-state solution. I assume so below.)

To do that, I must find a solution for $$\frac{dx}{a(x-1)+bx(x-1)+cx(y-1)}=\frac{dy}{a(y-1)+by(x-1)+cy(y-1)}.$$ How can I proceed?

(Background: $F(x,y,t)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}x^my^n P(m,n,t)$ for some probability distribution $P(m,n,t)$, where $m$ and $n$ are nonnegative integers. $F(1,1,t)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} P(m,n,t)=1$ for all $t$.)

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