Nonlinear ODE Abel equation of the first kind Edited: I have the following equation that represents a particular circuit. I am trying to get a solution that is practical for design purposes. I have went through some papers on Abel equations of the first kind, but I have not been able to get a reasonable solution. Thank you 

$\frac{dy}{dx} + A_1y + A_3y^3 = F(x)$

Ok, let's say $F(x)$ is sinusoidal. It can be any waveform but let's say it is $F\sin(x)$ where F is just an amplitude.

$\frac{dy}{dx} + A_1y + A_3y^3 = F\sin(x)$
Hint (Before edit): I assume $A_1$, $A_3$ and $F$ to be constants. Then this differential equation is separable.

$$\dfrac{dy}{F-A_1y-A_3y^3}=dx$$

The solution is not very nice. But if you can narrow the range of $y$ you could simplify it.



EDIT: In the case of a general $F(x)$ it is very likely that there is no simple closed form solution to the ODE. If you are trying to solve a practical problem you could try to look how much $F(x)$ does change in the practical range of values for $x$. It might be, that $F(x)\approx \text{const.}$ in the practical range of values.
Hint (Before edit): I assume $A_1$, $A_3$ and $F$ to be constants. Then this differential equation is separable.

$$\dfrac{dy}{F-A_1y-A_3y^3}=dx$$

The solution is not very nice. But if you can narrow the range of $y$ you could simplify it.



EDIT: In the case of a general $F(x)$ it is very likely that there is no simple closed form solution to the ODE. If you are trying to solve a practical problem you could try to look how much $F(x)$ does change in the practical range of values for $x$. It might be, that $F(x)\approx \text{const.}$ in the practical range of values.