I wonder if it is possible to solve analytically the following equation


$$
\dot{\alpha}\_t = -\frac{2}{m} \alpha^2\_t + \frac{1}{2m} (\alpha\_t - \alpha\_t^\*)^2
$$


Where $\alpha\_t$ is a complex function, $\alpha\_t^\*$ is its complex conjugate and $\dot{\alpha}\_t$ is the time derivative.


All the best!



Solution of this differential equation
Yes, this can be integrated explicitly. First, notice that, since $m\not=0$, we can write $\alpha(t) = 2m\bigl(x(t)+iy(t)\bigr)$, in which case, the given equation becomes
$$
\dot x + i\,\dot y = -4(x + iy)^2 + (2iy)^2 = -4 x^2 - i\,(8xy),
$$
so $\dot x = -4x^2$ and $\dot y = -8 xy$. Thus, by standard ODE techniques,
$$
x(t) = \frac{x(0)}{(1+4x(0)\,t)}\quad\text{and}\quad
y(t) = \frac{y(0)}{(1+4x(0)\,t)^2}\,.
$$


