TITLE: Solving differential equation $x''(t)=x^6$.
QUESTION [5 upvotes]: Solve the following differential equation
$$x''(t)=x^6(t)$$
If I had $x'(t)$ instead of $x''(t)$ the exercise would have been easier for me.
I would appreciate some help with this problem. 
Thank you very much.

REPLY [4 votes]: let $$y = \frac{dx}{dt}.$$  then we have $$\frac{dx}{dt} = y, \frac{dy}{dt} = x^6.$$ dividing one by the other gives $$ y\, dy - x^6 dx= 0$$ which has the integral $$y = \pm \sqrt{\frac27}\sqrt{k + x^7} \to  
 t=   \pm \sqrt{\frac72}\int_0^x \frac{dx}{\sqrt{k + x^7}}$$