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etotheipi
I've generally solved introductory second order differential equations the 'normal' way; that is, using the auxiliary equation, and if it is inhomogeneous looking at the complementary function as well, and so on.
I know that sometimes it can be helpful to propose an ansatz and substitute it into determine unknown coefficients. In this section of the Feynman lectures (http://www.feynmanlectures.caltech.edu/I_23.html) he proposes a solution to a damped oscillator with forced oscillations of the real part of $$x = \hat{x}e^{iwt}$$where the original differential equation was$$m \ddot{x} + c \dot{x} + kx = F_0 \cos{wt}$$How has he determined that this is a suitable ansatz for the scenario? Is this particular ansatz only applicable when forced oscillations are involved (excluding the obvious simplest case with no damping or periodic force)?N.B. I am not concerned about the complex exponential, the real part of which is [itex]\cos{\omega t}[/itex], but more why this expression has been deemed suitable for the case with forced oscillations (though of course it does actually work, and we could also obtain a solution of that form from first principles of differential equations).
Thank you!
I know that sometimes it can be helpful to propose an ansatz and substitute it into determine unknown coefficients. In this section of the Feynman lectures (http://www.feynmanlectures.caltech.edu/I_23.html) he proposes a solution to a damped oscillator with forced oscillations of the real part of $$x = \hat{x}e^{iwt}$$where the original differential equation was$$m \ddot{x} + c \dot{x} + kx = F_0 \cos{wt}$$How has he determined that this is a suitable ansatz for the scenario? Is this particular ansatz only applicable when forced oscillations are involved (excluding the obvious simplest case with no damping or periodic force)?N.B. I am not concerned about the complex exponential, the real part of which is [itex]\cos{\omega t}[/itex], but more why this expression has been deemed suitable for the case with forced oscillations (though of course it does actually work, and we could also obtain a solution of that form from first principles of differential equations).
Thank you!
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