TITLE: Showing that all solutions to the ODE $x''=4x^3$ cease to exist after a finite time
QUESTION [6 upvotes]: I am given the "Newton equation" $$x''(t)=4x^3(t)$$
The first part asks for finding a "conservation of energy" law. I'll spare the details:
the conclusion is that every solution has a constant $C$ such that $\frac{y^2}{2}-x^4=C$ where $y(t)=x'(t)$. (That follows from writing the ODE as a system of two ODEs: $x'=y$ and $y'=4x^3$ which is exact).
The second part asks to show that except the trivial solution $x\equiv0$, all solutions cease to exist (and therefore tend to infinity) after a finite time.
Now, this is something that can't be inferred by knowing the relation between $x$ and $y$ because it relies on their dependence on $t$, so I suspect I don't have the means to solve it.
I'd like to hear what is the technique for these questions so I can do the rest of the problems in my homework that contain similar questions (like, which solutions are bounded).
Thanks!

REPLY [4 votes]: Since $$y=\frac{dx}{dt}\\
t=\int_{x(0)}^\infty \frac{dx}y\\
=\int_{x(0)}^\infty \frac{dx}{\sqrt{2C+2x^4}}$$
and you need to show this is finite.