Finding the function when the second derivative is defined in terms of it.

Question

So I was playing around with derivatives and I realized that I could find a function when it's derivative was given in terms of it. But when I tried this with a second derivative I couldn't find the function.

Is there any way to solve for $f(x)$ in this equation? $$ \frac {d^2f}{dx^2} = (f(x))^{-2}$$

Answer 1

Hint: For differential equation of form $f''=F(f,f')$, the standard track is set $v=f'$, then we have $$ f''=\frac{\text dv}{\text dx}=\frac{\text dv}{\text df}\frac{\text df}{\text dx}=v\frac{\text dv}{\text df}, $$

then we get $$ v\frac{\text dv}{\text df}=f^{-2}. $$

Answer 2

$$y''=\frac 1 {y^2}$$ Multiply by $2y'$ $$2 y''y'=2\frac {y'}{y^2}$$

$$(y'^2)'=-2(y^{-1})'$$ Then integrate $$y'^2=K-2y^{-1}$$