Let's suppose I'm running in a circle. The maximum acceleration is limited by a value $\mu$:
$v^2/R = \mu*g$
So, if I'm running in that circle at a speed lower than the max speed (the lateral acceleration is lower than $\mu*g$), I can accelerate. The longitudinal acceleration is:
$a = ((\mu*g)^2 - v^4/R^2)^{0.5}$
If I accelerate, the speed increases, so there will be less longitudinal acceleration. I can write:
$dv/dt = ((\mu*g)^2 - v^4/R^2)^{0.5}$
How can I solve this? I need to remove $v$ from the square root, but how?
