What is the general solution of differential equation $y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$ What is the general solution of differential equation $y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$.
The answer to this DE is $log(y) = c_1 e^x + c_2 e^{-x}$
I don't know the method to solve differential equation with degree more than 1. Please tell me how to solve these types of equations.

Let's suppose $y\neq 0$, then the given DE is equivalent to
$$\dfrac{y\dfrac{d^2y}{dx^2}-\left(\dfrac{dy}{dx}\right)^2}{y^2}=\log y$$
i.e. $$\dfrac{d}{dx}\left(\dfrac1y\dfrac{dy}{dx}\right)=\log y$$
By making the sustitution $u=\log y$ the last DE becomes
$$\dfrac{d}{dx}\left(\dfrac{du}{dx}\right)=u\qquad \text{i.e.} \qquad u''-u=0$$
last equation is linear and homogeneous, its solution is given by $u=c_1e^x+c_2e^{-x}$.
