First order nonlinear non-exact ODE

Question

Please help me to find the solution of $$ (2x+y)dx+(xy+2)dy=0 $$ with $y$ as dependent variable.

I've tried to find the integrated factor, multiplied it with $x, y, xy, 1/x, 1/y, 1/xy$ and still couldn't find the solution. Please help.

Answer 1

Similiar to None exact first order ODE:

$(2x+y)~dx+(xy+2)~dy=0$

$(-2x-y)\dfrac{dx}{dy}=xy+2$

This belongs to an Abel equation of the second kind.

Let $u=-2x-y$ ,

Then $x=-\dfrac{u+y}{2}$

$\dfrac{dx}{dy}=-\dfrac{1}{2}\dfrac{du}{dy}-\dfrac{1}{2}$

$\therefore u\left(-\dfrac{1}{2}\dfrac{du}{dy}-\dfrac{1}{2}\right)=-\dfrac{y(u+y)}{2}+2$

$u\dfrac{du}{dy}+u=yu+y^2-4$

$u\dfrac{du}{dy}=(y-1)u+y^2-4$

Let $s=y-1$ ,

Then $\dfrac{du}{dy}=\dfrac{du}{ds}\dfrac{ds}{dy}=\dfrac{du}{ds}$

$\therefore u\dfrac{du}{ds}=su+(s+1)^2-4$

Let $t=\dfrac{s^2}{2}$ ,

Then $\dfrac{du}{ds}=\dfrac{du}{dt}\dfrac{dt}{ds}=s\dfrac{du}{dt}$

$\therefore su\dfrac{du}{dt}=su+(s+1)^2-4$

$u\dfrac{du}{dt}=u+s+2-\dfrac{3}{s}$

$u\dfrac{du}{dt}=u\pm\sqrt{2t}+2\mp\dfrac{3}{\sqrt{2t}}$

This belongs to an Abel equation of the second kind in the canonical form.

Please follow the method in https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf