# General solution of ODE in parametric form

by Adam Kaczmarek   Last Updated August 14, 2019 09:20 AM - source

I'm having trouble with obtaining parametric form of general solution solution of given ODE ($$A,\; B,\; C$$ are constants ):

$$A y''(t)+B y(t) y'(t)+ Cy(t)=0$$

This is autonomous differential equation which can be converted to 1st order equation using $$y'(t)=\omega (y)$$, which results in Abel equation of the second kind:

$$A\omega(y)\omega'(y)+B y \omega(y)+ Cy^3=0$$

I can obtain solution of this ODE using maple but result isn't in parametric form (t is explicite),and parametric form should look like this:

$$y=(-2A/B z)^{1/2}$$ where $$z=\eta Exp(- \int\frac{\sigma d\sigma}{\sigma^2-\sigma-D})$$ with $$D=-2AC/B^2$$ and $$\eta$$ is constant, $$y'=\sigma z$$

and

$$y''=-B/A\frac{\sigma+D}{\sigma}y y'$$.

This equation comes from https://arxiv.org/abs/1710.01910 (eq 81 with $$y(t)=H(t)$$ and solution: 83-85).

Here's my question: how to solve/obtain parametric form of this equation (or Abel one)? I've only found this paper http://real.mtak.hu/71378/1/1246.pdf but Abel equations in it are different than this given above

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