Can someone give me a hint to solve the next nonlinear diferential equation? \begin{equation} \frac{y'}{y}+\frac{f(t)+g(t)y}{h(t)+k(t)y}=0 \end{equation}
in some set where is well defined.
I know that if for example $k:=0$ this equation is reduced to a Bernoulli differential equation, and if $f=h$ and $g=k$ the soulution is $y=e^{-t}$. But I don't know how to solve this equation in a general case. Is there some way to solve it whitout using approximations when $f,h,g,k$, are constants, I mean \begin{equation} \frac{y'}{y}+\frac{f+gy}{h+ky}=0 \end{equation} I just have taken a ordinary differential equations course so I don't know too much about nonlinear differential equations so if someone could give me some hint or an advice I really appreaciate it. Sorry for my english I am new writing in enlgish.
Your equation is an Abel equation of the second kind, which does not have a simple general solution. Simply multiply by $h(t)+k(t)y$ to get it to a more common form, \begin{align} [h(t)+k(t)y]y'+g(t)y^2+f(t)y=0. \end{align} There are several transformations you can do to bring it to 'simpler' forms, but the solution for $y(t)$ is quite involved except for a select amount of specific cases. Here is a reference on the equation (you have to transform $h+ky=ku$ to get it to the form on the page), and papers such as this one show the method for the general solution.