During inflation, the expansion of the universe is governed by the Friedmann equation,

$$H^{2}=\frac{8 \pi G}{3 c^{2}}\left(\frac{1}{2} \dot{\phi}^{2}+V(\phi)\right)$$

and the equation of motion for the inflaton field $\phi$,

$$\ddot{\phi}+3 H \dot{\phi}+\frac{\partial V}{\partial \phi}=0 \text {. }$$

The slow-roll conditions are $\dot{\phi}^{2} \ll V(\phi)$ and $\ddot{\phi} \ll H \dot{\phi}$. Under these assumptions, solve for $\phi(t)$ and $a(t)$ for the potentials:

(i) $V(\phi)=\frac{1}{2} m^{2} \phi^{2}$ and

(ii) $V(\phi)=\frac{1}{4} \lambda \phi^{4}, \quad(\lambda>0)$.