Let $p,q$ be analytic functions in the unit disk $𝒰$. For $\alpha \in [0,1)$ the authors consider the differential subordination and the differential equation of the Briot-Bouquet type: $$p^{1-\alpha }\left(z\right)+\frac{z{p}^{\text{'}}\left(z\right)}{\delta p^{\alpha }\left(z\right)+\lambda p\left(z\right)}\prec h\left(z\right),z\in 𝒰,$$