Michael Fowler (closely following Landau para 22)
Consider a one-dimensional simple harmonic oscillator with a variable external force acting, so the equation of motion is
which would come from the Lagrangian
(Landau "derives" this as the leading order non-constant term in a time-dependent external potential.)
The general solution of the differential equation is , where , the solution of the homogeneous equation, and is some particular integral of the inhomogeneous equation.
An important case is that of a periodic driving force A trial solution
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