which is basically the same differential equation but with the right-hand-side set to zero. For such a homogeneous equation, if you have a solution, you can scale it by any constant and still get a solution. You can add it to any other solution and still get a solution.
You might see how this helps us get the general solution for our original inhomogeneous equation.
  1. Find the general solution \(x_h\left(t\right)\) for the associated homogeneous equation, which is what you did in the previous section (without the external driving force).
  2. Find a particular solution \(x_p\left(t\right)\) for the actual inhomogeneous equation.
  3. The general solution for the inhomogeneous equation is \(x\left(t\right)=x_h\left(t\right)+x_p\left(t\right)\). [It makes sense, and more rigorous proof is left to the interested reader.]

3.2 We have the solution for the associated homogeneous equation from the previous section (consider only the under-damped case). We just need to guess a particular solution. Try \(x_p=\alpha\cos\left(\Omega t\right)+\beta\sin\left(\Omega t\right)\), since we expect the system to move according to the external driving force (physically, this is certainly the expectation in the steady state, after any transient effects have been damped away). By making the appropriate substitutions into the inhomogeneous differential equation, obtain an equation that involves terms in \(\cos(\Omega t)\) and \(\sin(\Omega t)\).