# First order partial differential equation question

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.

Fix $y_0$ and define $v$ by $v(s)=u(s,y_0+s)$, then your differential equation reads $$v'(s)=-v(s)\cdot(L-y_0-s)^{-1},$$ solved by $v(s)=w(y_0)\cdot(L-y_0-s)$ for a given function $w$. Now $u(t,y)=v(s)$ for $s=t$ and $y_0=y-t$ hence $u(t,y)=w(y-t)\cdot(L-y)$.
The initial condition yields $w(-t)\cdot L=at$ hence $u(t,y)=(a/L)\cdot(t-y)\cdot(L-y)$.
Thank you, I'm quite sure I understand now. I have one final question though, I'm asked to sketch the graph of a typical curve in the (t, y) plane along which u(t, y) is constant, for example $u = au_{0}$. What am I sketching? The function u(t,y) with au_0 = u(t,y)? Thank you. – user16453 Nov 17 '11 at 11:59
You must draw the subset $C(u_0)$ of the $(t,y)$ plane of equation $u(t,y)=au_0$. For example $C(0)$ is the union of the lines $y=L$ and $t=y$. This is not the typical case though... – Did Nov 17 '11 at 19:40