 Solution

Set $t=0$ just inside the entrance, $z=0$, $\rho ={\rho }_0$ and $\theta ={\theta }_0$. The stated continuity gives $\dot{\rho }(0)=0$. Until the first axis crossing,For the full orbit, including any crossings, use the transverse complex coordinate $u=x+iy$:This helical solenoid trajectory with zero incident angular momentum is a circle of radius ${\rho }_0/2$ about transverse guiding centre $u_0/2$, combined with uniform axial motion. Its cyclotron angular frequency is $|qB|/(\gamma m)=2|\Omega |$, and its axial pitch per turn is $\pi w/|\Omega |$. The speed check is $|\dot{u}{|}^2+w^2={\Omega }^2{\rho }_0^2+w^2=v^2$.

The physical radius is $|u|={\rho }_0|\cos \Omega t|$. When the cosine changes sign, a nonnegative-radius coordinate adds $\pi$ to $\theta$; this is a coordinate change, not a discontinuous turn of the particle. For $q=0$ or $B=0$, the limit is simply $u=u_0$, $z=vt$.