Let us consider the following family of partial differential equations $$ \frac{\partial G(x,y,t)}{\partial t} = a(x) \frac{\partial G(x,y,t)}{\partial x} + b(y) \frac{\partial G(x,y,t)}{\partial y} + c(x,y) \frac{\partial G(x,0,t)}{\partial x} $$ If there was not the last term, this equation could be solved with the method of the characteristics. How can one deal with terms of the type $\frac{\partial G(x,0,t)}{\partial x}$?
For the specific problem I am studying, I am interested in the case $$ a(x) = b(x) = (1-x)( 1- \alpha x) $$ and $$ c(x,y) = \beta x(y-1) $$ where $\alpha$ and $\beta$ are parameters, and boundary condition $$ G(x,y,0) = x $$