Solving a nonlinear 1st order ODE

Question

I need help solving this nonlinear first order ODE. Any help will be appreciated.

$y'(x) + ay(x) + \frac{bx^2}{y(x)} + cx + d = 0$

Thank you.

Answer 1

Hint:

$y'(x)+ay(x)+\dfrac{bx^2}{y(x)}+cx+d=0$

$y\dfrac{dy}{dx}=-ay^2-(cx+d)y-bx^2$

This belongs to an Abel equation of the second kind.

Let $y=e^{-ax}u$ ,

Then $\dfrac{dy}{dx}=e^{-ax}\dfrac{du}{dx}-ae^{-ax}u$

$\therefore e^{-ax}u\left(e^{-ax}\dfrac{du}{dx}-ae^{-ax}u\right)=-ae^{-2ax}u^2-(cx+d)e^{-ax}u-bx^2$

$e^{-2ax}u\dfrac{du}{dx}-ae^{-2ax}u^2=-ae^{-2ax}u^2-(cx+d)e^{-ax}u-bx^2$

$e^{-2ax}u\dfrac{du}{dx}=-(cx+d)e^{-ax}u-bx^2$

$u\dfrac{du}{dx}=-(cx+d)e^{ax}u-bx^2e^{2ax}$