How to solve a Bernoulli ODE with an additional term? How to find the solution $y(t)$ of the following ODE  provided that:

*

*$y(t)$ has a strictly positive value at time $t=0$

*$\mu(t)$, $\alpha(t)$ and $\sigma(t)$ are known functions of time with all "suitable" properties.

*$y(t)$ has a strictly positive value at time $t=0$?

$$ \partial_{t}y(t)=\sigma(t)-\alpha(t) y\left(t\right)-\mu\left(t\right)y(t)^{2}$$
May be finding a solution for a more simple problem may help. For example, i study the case where both $\sigma(t)$ and $\alpha(t)$ are constant but not $\mu(t)$
Without the term $\sigma(t)$, it looks like a Bernoulli ODE but how to deal with this additional term?
 A: $$ \frac{\partial y}{\partial t}=\sigma(t)-\alpha(t) y\left(t\right)-\mu\left(t\right)y(t)^{2} \tag 1$$
It is common to solve a Riccati ODE if a particular solution is known. In the present case no particular solution can be found by inspection. So, in this case it is of interest to transform the non-linear first order ODE into a linear second order ODE because  a linear ODE is generaly easier to solve than a non-linear ODE even if the transformation increases the order of the ODE.
The usual method involves the change of function :
$$y(t)=\frac{u'(t)}{\mu(t)u(t)}$$
$$y'=-\frac{u''}{\mu u}-\frac{\mu'u'}{\mu^2u}-\frac{u'^2}{\mu u^2}=\sigma-\alpha\frac{u'}{\mu u}-\mu\left(\frac{u'}{\mu u} \right)^2$$
After simplification :
$$-\frac{u''}{\mu u}-\frac{\mu'u'}{\mu^2u}=\sigma-\alpha\frac{u'}{\mu u}$$
$$u''+(\frac{\mu'}{\mu}-\alpha)u'+\sigma\mu \:u=0$$
Let $\quad \frac{\mu'(t)}{\mu(t)}-\alpha(t)=f(t)\quad$ and $\quad\sigma(t)\mu(t)=g(t)$
$$u''(t)+f(t)u'(t)+g(t)u(t)=0$$
This is the general second order linear ODE. There is no standard special function available to express the solutions in the general case. Only for a few kind of functions $f(t)$ and $g(t)$ the solution can be explicitly expressed thanks to a limited number of elementary and available special functions.
Thus don't expect an explicit analytical solution of your equation $(1)$ for any functions $\sigma(t)\:,\:\alpha(t)\:,\:\mu(t)$ .
Even in the case $\sigma=$any constant and $\alpha=$any constant and $\mu(t)=$ any not constant function, the functions $f(t)$ and $g(t)$ remain any related functions. The conclusion remains the same : Don't expect a explicit solution with a finite number of standard and special functions, except for a few kind of functions $\mu(t)$.
The function $\mu(t)$ must be specified in the wording of the question if you want to know if the solution of Eq.$(1)$ can be or not explicitly expressed.
For example if $\mu(t)=\frac{c_1}{t+c_2}$ and $\sigma$ , $\alpha=$ constants the solution $y(t)$ involves the "confluent hypergeometric" functions.
