# Inflation problem for exponential potential

I'm trying to solve the inflation problem for exponential potential. $$v(\phi) = v_0 \exp(-\alpha \phi)$$ (it's known as barrow or power law inflation) we have two main equations: $$H^2 = 8π G / 3 (1/2 (\dot{\phi})^2 + v(\phi))$$ $$\ddot{\phi} + 3H \dot{\phi} + v(\phi)'=0$$ I must solve these two equations and find $\phi(t)$ & H(Hubble). in the book of cosmology by Weinberg has written,it is easy.many articles have mentioned it;but I can't do it.it has exact solution.answer is: $$\phi(t)=\phi_0 \ln t/ \ln t_0$$ and $$R(t)=R_0 (t/t_0)^b$$ with suitable constant b,$\phi_0$,$R_0$ .

Should I use slow-roll condition for solve that? ($\dot{\phi}^2 \ll v(\phi)$ and $\ddot{\phi} \ll v(\phi)'$) or without that I can?

$3H\dot{\phi} + V'(\phi)=0$
$H^2 = 8\pi G V(\phi) /3$
This slow roll approximation is valid if the slow roll parameters are $\ll 1$ which requires $\alpha \ll 1/m_p$.