TITLE: How to solve the differential equation $xx'' = (x')^2$?
QUESTION [6 upvotes]: There is an differential given:
$$xx'' = (x')^2,$$
where $x' \neq 0.$
I tried to solve this problem but I cannot see the proper substitution.

REPLY [5 votes]: $$xx'' = (x')^2,$$
Divide by $x^2 \,\, x\neq 0)$
$$\frac {x''}x = \frac {(x')^2}{x^2}$$
Note that 
$$(\frac {x'}x)'= \frac {x''}x-\frac {x'^2}{x^2}$$
Therefore
$$\frac {x''}x = \frac {(x')^2}{x^2} \implies (\frac {x'}x)' =0$$
$$\frac {x'}x =K_1 \implies \ln|x|=K_1t+K_2 \implies x=K_2e^{K_1t} $$