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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.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?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.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".