I'm sorry if this is a rather silly question, but it keeps bothering me that I can't find a satisying answer.

In Orbital Dynamics, we can describe the path taken by the body with the path equation: $$ \frac{d^2u}{d\theta^2} + u = - \frac{f(\frac{1}{u})}{L^2u^2}$$ $ u = \frac{1}{r(\theta)} $, and the force field $ F = mf(r)\hat{r} $.

However we can't solve this differential equation with the initial condition that the angular momentum is zero, i.e. the body is falling radially inwards, why isn't this orbit included in the differential equation?