I am currently working on problem in my own research. There seems to be a weak analogy between my problem and motion on a spring. Therefore, I am exploring this question in regards to a mass oscillating on a spring in hopes to gain further insight into my own system in question.
Here is the idea: We can write out the differential equation of motion for a mass on a spring
Even though we can find an analytic solution to this equation, let's assume that we can only solve this equation numerically.
Let's say we start the mass at rest at some non-zero initial position. We know that the amplitude of the oscillation will be equal to the magnitude of the initial position (for example, if we start at x = 5 m, then the amplitude will be 5 m).
My question is this: Is there a way to determine that this is true of the amplitude without actually solving the differential equation. In other words, can we use the equation (and maybe other things we know about the system, like how at the maximum position the velocity is 0 and the acceleration will be at a maximum) to determine what will be our maximum position without actually solving the differential equation.
I know this is kind of vague, so if more information is needed please let me know.