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Equation of motion, functions

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A simple harmonic oscillator, of mass m and natural frequency w_0, experiences an oscillating driving force f(t)=m a cos(wt). Therefore, its equation of motion is

d^2x/dt^2 + w^2_0x = acos(wt)

where x is the position. Given that at t=0 we have x=dx/dt=0, find the function x(t). Use BOTH with variation of the parameters and guessing methods. Describe the solution if w is approximately, but not exactly, equal to w_0. Give an example of a physical system where this happens.

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The equation is:
(1.1)
The general solution is given by:
(1.2)
Where is the solution of the associated homogenous equation
(1.3)
and is a particular solution to equation (1.1).
This equation is a second order linear equation with constant coefficients.
To solve equation (1.3) we "guess" a solution in the form:
(1.4)

Plugging it in the homogenous solution we get:

(1.5)
Where
Thus the two homogenous solutions are:
(1.6)
Since we are interested in real solutions, and the homogenous solution is a linear combination of these two linearly solutions, we can use Euler's identity
(1.7)
To express the two ...

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