Does this ODE have an exact or well-established approximate analytical solution? The equation looks like this:
$$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$
where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer.  Actually I am mainly concerned with the $n=4$ case.
The $n=1$ case is trivial.  Otherwise the only method I can think of is kind of an iterative approximation, in which a "clean" expression seems not easy to obtain.
Just in case it is too "easy" for the experts, we may generalize it to the form
$$\frac{\mathrm{d}y}{\mathrm{d}t} = f(t) - g(y).$$

You may be interested in the following power series solution for the case $n=4$
$$ y \left( t \right) = y \left( 0 \right) + \left( A-C \left( y\left( 0
 \right)\right) ^{4} \right) t+ \left( \frac{1}{2}\,B\omega-2\,C \left( y
 \left( 0 \right)  \right) ^{3}A+2\,{C}^{2} \left( y \left( 0 \right) 
 \right) ^{7} \right) {t}^{2}+O \left( {t}^{3} \right).  $$
