 Bernoulli stochastic differential equation

On the positive half-line, divide by the stochastic exponential $F_t=e^{\alpha B_t-{\alpha }^{2}t/2}$. The Itô product rule reduces the equation to the random Bernoulli differential equation $d(X/F)/dt=F^{\delta -1}(X/F{)}^{\delta }$. For $\delta \ne 1$ this gives $X_t=F_t[x_0^{1-\delta }+(1-\delta ){\int }_0^t{F}_s^{\delta -1}ds{]}^{1/(1-\delta )}$, until the bracket vanishes; for $\delta =1$, $X_t=x_0{F}_t{e}^t$.