Solving non-linear ordinary differential equation: $(y'')^2-y'y'''=\left(\frac{y'}{x}\right)^2$
The most obvious thing to do at the moment is to start by substituting $v=y'$: $$(v')^2-v\cdot v''=\frac{v^2}{x^2}$$ Let's divide both sides by $v^2$, then integrate both sides with respect to $x$:
$$\int \frac{(v')^2-v\cdot v''}{v^2}=\int \frac{1}{x^2}~dx$$ It is easy to see via the quotient rule that: $$\left(\frac{v'}{v}\right)'=\frac{v\cdot v''-(v')^2}{v^2}$$ Therefore, we obtain a first-order separable ODE, which is easy to solve: $$-\frac{v'}{v}=-\frac{1}{x}+C \iff \frac{v'}{v}=\frac{1}{x}+k$$ Where $k:=-C$. All that remains to do is to solve for $v(x)$ explicitly, substitute back for $v=y'$ and integrate both sides with respect to $x$. Doing this will give you an explicit solution for $y(x)$.
Solving :
$$ \Bigg(\frac{d^2 y(x)}{dx^2}\Bigg) - \frac{dy(x)}{dx}\frac{d^3 y(x)}{ dx^3} = \frac{\frac{dy(x)}{dx}^2}{x^2}$$
Let :
$$\frac{dy(x)}{dx} = v(x)$$
which yields :
$$\frac{d^2 y(x)}{dx^2} = \frac{dv(x)}{dx}$$
$$\frac{d^3 y(x)}{ dx^3}= \frac{d^2 v(x)}{dx^2}$$
You get :
$$\Bigg( \frac{dv(x)}{dx}\Bigg)^2 -v(x)\frac{d^2 v(x)}{dx^2} = \frac{v(x)^2}{x^2}$$
$$\Rightarrow$$
$$\Bigg[-x^2 \Bigg( \frac{dv(x)}{dx}\Bigg)^2 + v(x)^2 + x^2\frac{d^2 v(x)}{dx^2}v(x)\Bigg]\frac{1}{x^2} = 0 $$
$$\Rightarrow$$
$$\frac{1}{x^2} + \frac{\frac{d^2 v(x)}{dx^2}}{v(x)} - \frac{\Bigg(\frac{dv(x)}{dx}\Bigg)^2}{v(x)^2} = 0 $$
$$\Rightarrow$$
$$\int \Bigg[\frac{1}{x^2} + \frac{\frac{d^2 v(x)}{dx^2}}{v(x)} - \frac{\Bigg(\frac{dv(x)}{dx}\Bigg)^2}{v(x)^2}\Bigg]dx = \int0dx$$
$$\Rightarrow$$
$$- \frac{1}{x} + \frac{\frac{dv(x)}{dx}}{v(x)} = c_1$$
$$\Rightarrow$$
$$\frac{dv(x)}{dx} = \frac{(c_1x+1)v(x)}{x}$$
$$\Rightarrow$$
$$\int \frac{dv(x)}{dx} dx = \int \frac{(c_1x+1)v(x)}{x} dx$$
$$\Rightarrow$$
$$\ln{v(x)} = \ln x + c_1x + c_2$$
$$\Rightarrow$$
$$v(x) = e^{c_1x + c_2}x$$
$$\Rightarrow$$
$$v(x) = c_2 c_1^x x$$
Substituting back for $\frac{dy(x)}{dx} = v(x)$, you get :
$$\frac{dy(x)}{dx} = c_2 c_1^x x $$
$$\Rightarrow$$
$$y(x) = \int c_2 c_1^x x dx = \frac{c_2 (\ln{(c_1)}x-1)c_1^x}{\ln^2c_1} + c_3 $$