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I understand the rest of your argument, but this was my original question and i still not getting why this is a possible solution, i mean, i can´t made a proper demostration or something. Is like I'm not taking into account the constants that integrals implies.Freaky Fred said:It was considered that the solution is of the type: x = ert
This is a strategy to solve this kind of second order linear equation.
Are you asking where ##x=e^{rt}## came from?velvetmist said:I understand the rest of your argument, but this was my original question and i still not getting why this is a possible solution, i mean, i can´t made a proper demostration or something. Is like I'm not taking into account the constants that integrals implies.
Thank you so much! I feel pretty silly now tbqh.Nugatory said:Are you asking where ##x=e^{rt}## came from?
It's an educated guess as to the form of the solution. You look at ##\frac{d^2x}{dx^2}+\frac{k}{m}x=0##, you see that this can only work if the second time derivative of ##x## is a multiple of ##x##, you remember that ##x=e^{rt}## has this property so you try substituting that into the equation and see if it works. Differential equations are solved this way so often that there's even a word for the initial educated guess as to the form of the solution: "ansatz".
The equation x=e^(rt) is derived from the differential equation for simple harmonic motion, which describes the motion of an object under the influence of a restoring force that is proportional to its displacement from equilibrium.
The equation x=e^(rt) represents the displacement of an object undergoing simple harmonic motion at any given time t. It shows that the displacement is directly proportional to the exponential function of time, with the constant r representing the frequency of oscillation.
The exponential function is used to represent the behavior of the displacement in simple harmonic motion because it is the solution to the differential equation for this type of motion. It accurately describes the oscillatory behavior of the displacement over time.
The variable r represents the angular frequency of the oscillations in simple harmonic motion. It is related to the period and frequency of the motion by the equations T=2π/r and f=1/T.
Yes, the equation x=e^(rt) can be applied to all types of simple harmonic motion, as long as the motion is described by a differential equation of the form x''+r^2x=0, where x is the displacement and r is a constant. This includes pendulum motion, spring motion, and other types of oscillatory motion.