Yes, you get a 1st order ODE.

Some hints...

We have

$\displaystyle \dot{\phi} = \frac{2}{R} \sqrt{\frac{(M-\mu m_B gR)\phi}{(3m_A + 2m_B)}}$

If we let

$\displaystyle A = \frac{2}{R} \sqrt{\frac{(M-\mu m_B gR)}{(3m_A + 2m_B)}}$

we can reduce the formula to

$\displaystyle \dot{\phi} = A\phi^{1/2}$

This makes the ensuing algebra a bit more manageable. Then you can then perform another substitution of the form

$\displaystyle u = \phi^{1/2}$

$\displaystyle \frac{du}{d \phi} = \frac{d\left(\phi^{1/2}\right)}{d\phi} = \frac{1}{2}\phi^{-1/2} = \frac{1}{2u}$

and form a chain rule to describe the ODE in terms of $\displaystyle u$ and $\displaystyle \dot{u}$. That ODE will then be much easier to solve.

Have a go

Give us a shout if you get stuck.