This is for mechanics II.
We're working with a particle of mass \(m\) in one-dimensional motion.
The potential energy is given by
\(V(x)=-\frac12kx^2\)
and the force is anti-restoring at
\(F(x)=kx\)
It is also given that we see unstable equilibrium at \(x=0\).
We want to consider the initial conditions
\(\quad\)\(t=0\\x=x_0\\\dot x=0\)
And we want to show that the motion is an "exponential 'runaway'"
\(x(t)=\frac12x_0(e^{\alpha t}+e^{-\alpha t})\)
where \(\alpha=\sqrt{\frac km~~}\)

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I know I should use \(F(x)=m\ddot x\) somwhere...

somewhere*

The \(e^{stuff}\) will probably come from integration with \(x\)'s in the denominator.

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