To get a feel for how the solution for the driven, damped oscillator was developed, let's do an easier problem using the same process. A particle of mass \( m \) is subject to a driving force \( F(t)=m a_{0} \mathrm{e}^{-\xi t} \) where \( a_{0} \) and \( \xi \) are constants. No spring is present. No damping is present. The initial position and speed are both zero. Find \( x(t) \), starting from the governing differential equation derived from Newton's Second Law:
\[
m \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=F(t) \quad \Longrightarrow \quad m \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=m a_{0} \mathrm{e}^{-\xi t} \quad \Longrightarrow \quad \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=a_{0} \mathrm{e}^{-\xi t}
\]
Here's a recipe for how I'd like you to find \( x(t) \) :
(a) Solve the homogeneous equation (with \( F=0 \) ). Yes, it's really simple. You can just state the result for \( x_{\mathrm{h}}(t) \). It will have two constants of integration, but you do not yet know the meaning of the integration constants.
(b) Now guess a solution for the particular solution. Take a cue from the derivation done in class and guess a form related to the driving force.
(c) Set up the auxiliary equation for the particular solution \( x_{\mathrm{p}}(t) \) to find the exact form of the particular solution.
(d) Now that you have the homogeneous and particular solutions, you should be able to write down \( x(t) \). Now you can apply the initial conditions to find the two integration constants.
(e) Setting \( a_{0}=1 \) and \( \xi=1 \), plot your solution for \( x(t) \) for \( t \) in the range from 0 to 3 .
