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The general trick is to convert the one second-order differential into two first-order ones. This is done by introducing an extra variable to carry around:
\begin{align}
\frac{\mathrm dv}{\mathrm dt}&=f(x,\,t)\\
\frac{\mathrm dx}{\mathrm dt}&=v
\end{align}
for whatever function $f(\cdot)$ you need.
This can easily be applied to numerical integration ...
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Both approaches are actually the same, if you do them correctly.
I will address your second case first. You are correct to use conservation of energy and say that the potential energy stored in the spring at the lowest point is equal to the sum of the kinetic energy and the potential energy due to gravity at the equilibrium point. So you were correct with ...
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