TITLE: ODE solution existence and uniqueness
QUESTION [0 upvotes]: my question is: can i apply the theorem I mentioned in the title to assert that there is only a solution(given initial conditions) to a energy conservation equation?
I'm talking about an equation like this:
$ E= m \dot x^2 + kx $
My doubts come from the fact that the equation isn't linear, but I can obtain a form like this:
$ \dot x = \pm \sqrt{ f(x)} $
So we have two equations that singularly seem to respect the hypotesis of the theorem cited, but I'm not sure they do that when together.

REPLY [0 votes]: The equation
$$
2E=m\dot x^2+kx^2
$$
can be locally uniquely solved for $\dot x$ if $\dot x\ne 0$. This local explicit form is then also continuously differentiable under this restriction. This gives monotonic solution segments.
It also has constant solutions at $\dot x=0$, that is, at $x=\pm\sqrt{2E/k}$. At these constant solutions there is no Lipschitz condition, the derivative is unbounded around it. More, one can easily construct piecewise solutions that alternate monotonous and constant segments.
Only by requiring that the second derivative be continuous one gets the "physical" solutions.
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