Non-linear second order equation with the derivative term

Question

I am trying to solve the following equation:

$y''(x) - \beta H(x) y'(x)=- (2 p - 2) y' ^ {2}(x)/y(x)$

where p and $\beta$ are constant and H is a function of x. Please help me with this equation.

Answer 1

Dividing by $y'$ we get $$\frac{y''}{y'} -\beta H =(2-2p)\frac{y'}{y}$$

integrating with respect to $x$ we obtain

$$\ln |y'| +(2p-2)\ln |y| =\beta \int H dx $$

$$\ln \left|\left(\frac{y^{2p-1}}{2p-1}\right)' \right|=\beta \int H dx $$ $$\left(\frac{y^{2p-1}}{2p-1}\right)' =e^{\beta \int H dx }$$ $$\frac{y^{2p-1}}{2p-1}=\int \left(e^{\beta \int H dx}\right)dx$$