Confusion...

Q) A particle is moving in a circle of radius R, such that at every instant the magnitudes of the instananeous accn and the radial accn are equal. If vel. of the particle be v₀ at time t=0, find the time reqd for the completion of half of the 1st revolution.

What I did was this :-

\frac{v^2}{r}=\frac{dv}{dt}=\frac{vdv}{dx}

Integrating we have v=e^{\pi}.v_0.

Now applying v=\frac{dx}{dt}=v_0.e^{\pi} & \int_{0}^{\pi r}{dx}=\int_{0}^{T}{v_0.e^{\pi}dt}, I'm getting T=\pi \frac{R}{v_0}e^{-\pi}......which surely is one of the options but sadly - not the ans.

Where am I goin' wrong?