The von Foerster equation is a continuity equation; it can be solved using the method of characteristics. Another way is by similarity solution; and a third is a numerical approach such as finite differences.
To get the solution, the following boundary conditions should be added:
which states that the initial births should be conserved (see Sharpe–Lotka–McKendrick’s equation for otherwise), and that:
which states that the initial population must be given; then it will evolve according to the partial differential equation.