Problem 1 of section 2.7 of Arnold's Ordinary Differential Equations book asks to prove that the period of the oscillations in the Lotka-Volterra model tends to infinity as the initial condition $(x_0,y_0)$ tends to $(0,0)$. The exposition is as follows: one begins with the Lotka-Volterra system
$$\dot x = kx-axy=(k-ay)x\qquad \dot y = -\ell y + bxy=(bx-\ell)y$$
Then one notices that this corresponds to a vector field in the phase space. One may rescale the vector field at each point $(x,y)$ to get a new system of ODEs
$$\dot x = \frac{x}{bx-\ell}\qquad \dot y = \frac{y}{k-ay}$$
and this system will have the same phase curves, up to re-parametrization. Arnol'd then works with the latter system a bit to show that the solution curves are closed, but he never explicitly solves anything.
Here is the problem. I know that the period of the oscillations for the latter system is $T=bx_0 + t_0$. How do I relate this to the period of the oscillations of the former system, since the solution curves are reparametrized when passing from the former to the latter?