How to solve the differential equation $xx'' = (x')^2$?

Think that

$$ \frac{\ddot x}{\dot x} = \frac{\dot x}{x}\Rightarrow \frac{d}{dt}\ln(\dot x) = \frac{d}{dt}\ln (x) $$

so

$$ \ln(\dot x) = \ln(x) + C \Rightarrow \dot x = C_1 x\Rightarrow x = C_2e^{C_1 t} $$

The comment seen here uses a "trick". Here is a more standard solution.

The independent variable is not rendered explicitly so you have an autonomous equation. Given this, put in $u=x'$. Then

$$x''=\frac{d}{dt}u(x(t))=\frac{du}{dx}\, x'=u\,\frac{du}{dx}$$

using the Chain Rule. So

$$x\cdot u\cdot \frac{du}{dx}=u^2$$

Divide by $u$ and solve the separable equation that results, then substitute $x'$ for $u$ into that first integration to get another first order equation that is easily solved.

With the substitution $$v(x)=\frac{dx}{dt},$$ we get $$x\frac{dv(x)}{dx}v(x)=(v(x))^2.$$ This can be written as $$-v(x)\left(-x\frac{dv(x)}{dx}+v(x)\right)=0,$$ so either $v(x)=0$ or $$\frac{dv(x)}{dx}=\frac{v(x)}{x}$$ and this is $$\int\frac{\frac{dv(x)}{dx}}{v(x)}dx=\int\frac{1}{x}dx.$$ Can you finish?