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## Homework Statement

The harmonic oscillator's equation of motion is:

x'' + 2βx' + ω

_{0}

^{2}x = f

with the forcing of the form f(t) = f

_{0}sin(ωt)

## The Attempt at a Solution

So I got:

X

_{1}= x

X

_{1}' = x' = X

_{2}

X

_{2}= x'

X

_{2}' = x''

∴ X

_{2}' = -2βX

_{2}- ω

_{0}

^{2}X

_{1}+ sin(ωt)

The function f(t) is making me doubt this answer because I have to take into account f

_{0}and it just disappears in the solution.

(For context I have to put it into a system of first order ODEs because I have to code it into python and plot the results with the given parameters.)

Am I on the right track? Or am I missing anything?