Second order ODE into a system of first order ODEs

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The discussion revolves around converting the second-order ordinary differential equation (ODE) of a harmonic oscillator into a system of first-order ODEs. The user correctly identifies the variables X1 and X2 to represent position and velocity, respectively, but expresses concern about the forcing function f(t) and its impact on the solution. A participant points out that the forcing term seems to have been omitted in the user's equations, suggesting it should not be disregarded. The user acknowledges this oversight and considers the need for initial conditions in their solution. The conversation emphasizes the importance of including all terms in the equations when transitioning between ODE forms.
whatisgoingon
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Homework Statement


The harmonic oscillator's equation of motion is:

x'' + 2βx' + ω02x = f

with the forcing of the form f(t) = f0sin(ωt)

The Attempt at a Solution



So I got:
X1 = x
X1' = x' = X2
X2 = x'
X2' = x''
∴ X2' = -2βX2 - ω02X1 + sin(ωt)

The function f(t) is making me doubt this answer because I have to take into account f0 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?
 
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whatisgoingon said:
it just disappears in the solution
Looks to me you made it disappear in the equations already.
 
BvU said:
Looks to me you made it disappear in the equations already.
Oh so I shouldn't have taken it out in the first place then? That makes sense thanks! Other than that, am I missing anything else?
 
Initial conditions ?
 

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