I came across this differential equation which I'm having trouble finding an analytic solution to:


I'm trying to solve for y. I have initial conditions as $x_0=0.02$ and $y_0=100000$, and A and B are known constants. I don't have a very heavy differential equations background so all I know is that I can't use separation of variables--what kind of method should I use to solve this equation?

Thank you!!

  • 2
    $\begingroup$ You can use a numerical method! $\endgroup$ – Dr. Sonnhard Graubner Aug 1 '19 at 14:15
  • $\begingroup$ Try something along the lines of using $z = \frac{1}{x y}$ as a new independent variable... $\endgroup$ – vonbrand Aug 1 '19 at 18:16

Assume $A,B\neq0$ for the key case:


Let $u=xy$ ,

Then $y=\dfrac{u}{x}$






Let $v=x^2$ ,

Then $\dfrac{dv}{du}=2x\dfrac{dx}{du}$




This belongs to an Abel equation of the second kind.


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