The population density $u$ of a species at age $y$ and time $t$ is given by the equation
$u_{t} + u_{y} = \frac{-u}{(L-y)}$
$t\geq 0$ and $0 \leq y < L$
Initial conditions are $y = 0, u(t,0) = at$ for constant a $a > 0$
I'm trying to solve the equation using the method of characteristics but having no luck. So far this is what I've got:
$t = t_{0} + s$
$y = y_{0} + s$
$\frac{du}{ds} = \frac{-1}{(L-y)}$
I'm stuck and don't know what to do with the final characteristic equation and how to tie in the first two that I integrated. I know that the third equation is supposed to be an arbitrary function of the first two but I don't know what to do next.
The next two questions are asking me to sketch the graph of a typical curve in the (t,y) plane along which u(t,y) is constant, say $u = au_{0}$, which I'm not sure how to do either.
Thanks for reading.