Show an ODE has a global solution How do I show the following ODEs have global solution?


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*$x''+x+x^3=0$

*$x''+x'+x+x^3=0$

*$\begin{cases}x'=4xy^3+2x\\
      y'=-4x^3-2y-\cos(x)\end{cases}$

 A: Intuition from physics often helps with ODE. The first problem describes a 1D particle subject to the force $-x-x^3$ (assuming unit mass of the particle). The force is conservative (as are all 1D forces that depend only on position). Its potential function is $U(x)=x^2/2+x^4/4$. Therefore,  the total energy 
$$\frac12 (x')^2+U(x) \tag1$$
should be constant in time. Which you can check just by differentiating (1) in $t$; this does not rely on any facts from physics. Once you know that $x$ and $x'$ are uniformly bounded, the  Picard–Lindelöf theorem gives you an interval of existence of uniform size, which implies global existence.
Problem 2 differs from 1  only by $x'$, which in physical terms means dissipative force $-x'$. Now the function (1) will be decreasing in time instead of being constant. This again implies  that $x$ and $x'$ are uniformly bounded, and the rest goes as in 1. 
I don't see an easy approach to the last problem -- are you sure it's stated correctly? In any case, two out of three ain't bad.
