Determining ansatz for forced oscillations in SHM

[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 $\cos{\omega t}$, 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!

 

[member 428835]

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 $\cos{\omega t}$, 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!

Not entirely sure I understand your question, but I believe mathematicians looked at the linear ODE structure and thought what function, if differentiated, can be summed to be itself? The answer is, of course, a series solution taking the form $\sum_i a_i x^i$ where the coefficients $a$ are to be determined. This makes sense, right? Consider the homogenous solution to the undamped system $\ddot x + k/m x = 0$. Guessing $$y = \sum_ia_i x^i = a_0 + a_1x + a_2x^2+a_3x^3+... \implies\\
y''=2a_2+6a_3x+... \implies\\
\ddot x + \frac k m x = 0 \implies\\
2a_2+6a_3x+... + \frac k m (a_0 + a_1x + a_2x^2+a_3x^3+...) \implies\\
(2a_2+\frac k m a_0) + (6a_3 + a_1)x + ... = 0 + 0x \implies\\
2a_2+\frac k m a_0 = 0,\,\,\, 6a_3 + a_1 = 0$$

(sorry, my $x$'s should be $t$'s and my $y$'s should be $x$'s, but you get the idea.