Solution of generating function does not make sense

Question

Consider the generating function $$G(x,t) = \sum_{n=0}^N P_n(t) x^n,$$ with $G(1,t) = 1$ and $G(x,0) = x^m$. From a master equation, I obtained the following partial differential equation for $G$: $$\frac{\partial G}{\partial t} = (x-1) \left[2 a N G - (2a x + b) \frac{\partial G}{\partial x} \right].$$ This seems ripe for the method of characteristics and if we consider $G(x,t) = G(x(t),t)$ we obtain the characteristics $$C e^{(2a + b)t} = \frac{x-1}{2a x + b},$$ along which $G$ evolves $$G(x,t) = K[1 - 2 a C e^{(2a + b)t}]^{-N}.$$ The boundary condition $G(1,t)$ implies that along that characteristic, we must have $C = 0$ and so $K = 1$ (since $G(1,t) = 1$). At this point, I am a little confused since $$G(x,t) = [1 - 2 a C e^{(2a+b)t}]^{-N}$$ Implies with the characteristic that $$G(x,t) = G(x) = \left(\frac{2ax+b}{2a+b}\right)^N.$$ This is the generating function for the binomial distribution with $N$ trials and $p = \frac{2a}{2a+b}$, however there seems to be no time dependence on this solution (e.g., if I solve the PDE at steady state, I would have arrived at the same result), how is this possible?