Second order ODE into a system of first order ODEs

[whatisgoingon]

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?

 

[BvU]

it just disappears in the solution

Looks to me you made it disappear in the equations already.

 

[whatisgoingon]

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?

 

[BvU]

Initial conditions ?