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

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.

• $(x/x')'=((x')^2-xx'')/(x')^2$ may help. – Gerry Myerson May 11 '18 at 9:43
• Worth noting the next time you see a problem like this is that the closed-form solutions to second-degree differential equations of one variable often involve $c_1 \sin k_1 x + c_2 \cos k_2 x$, as the first and second derivatives have the same form. The $\sinh$ and $\cosh$ functions also often appear, for similar reasons. – Davislor May 16 '18 at 1:41

Think that

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

so

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

• Thanks LutzL for the hint. – Cesareo May 11 '18 at 10:59

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.

• Which works in this case because $u = x' \neq 0$. – Davislor May 16 '18 at 1:49
• Agreed. Ironically, the solution gotten by assuming $u\ne 0$ also covers $u=0$ given the proper value of one integration constant. Dumb luck strikes again! – Oscar Lanzi May 16 '18 at 1:55

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?

$$xx'' = (x')^2,$$ 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 $$\frac {x''}x = \frac {(x')^2}{x^2} \implies (\frac {x'}x)' =0$$ $$\frac {x'}x =K_1 \implies \ln|x|=K_1t+K_2 \implies x=K_2e^{K_1t}$$