Hmm, I must admit that I agree with you there. Before now I had just thought of this as 'standard' harmonic oscillator question, which of course it is not. The problem would also be straight forward if the damping term was velocity dependent, unfortunately, this is not the case, nor is the inhomogeneous term constant.
I'll also admit that the equation that we were originally working with is entirely incorrect as you point out. I should have read the OP fully.
So, let's take your proposed equation of motion (with the sign function) and see where we get:
x^{\prime\prime} + \frac{b}{m}\;\text{sign}\;x^\prime + \frac{k}{m}x = 0
Solving ODE's with the sign function is messy, so I propose that we explicitly split the EOM into the three distinct regions:
\left.\;\;\; \begin{aligned} x^{\prime\prime} - \frac{b}{m} + \frac{k}{m}x = 0 \;\;\;,\;\;\; x^\prime > 0 \\<br />
x^{\prime\prime} + \frac{k}{m}x = 0 \;\;\;,\;\;\; x^\prime = 0 \\<br />
x^{\prime\prime} + \frac{b}{m} + \frac{k}{m}x = 0 \;\;\;,\;\;\; x^\prime < 0 \end{array}\right\}
The equations in all three regions have the same homogeneous equation, with the corresponding indicial equation:
\lambda^2 + \frac{k}{m} = 0
Hence, the homogeneous solution can be written:
x = A^\prime\cos\left(\sqrt{\frac{k}{m}}t\right) + B\sin\left(\sqrt{\frac{k}{m}}t\right)
The particular integrals corresponding to regions one and three are simply additive constants (\mp \frac{b}{k}[/tex] respectively). For brevity, we recast the homogeneous solution exclusively in terms of cosine, with an arbitrary phase shift \phi. Hence, the solution in equation region may be written thus:<br />
<br />
\left.\;\;\; \begin{aligned} x = A\cos\left(\omega t -\phi\right)- \frac{b}{k} \;\;\;,\;\;\; x^\prime &gt; 0 \\<br />
x = A\cos\left(\omega t -\phi\right) \;\;\;,\;\;\; x^\prime = 0 \\<br />
x = A\cos\left(\omega t -\phi\right) + \frac{b}{k} \;\;\;,\;\;\; x^\prime &lt; 0 \end{array}\right\}.<br />
<br />
With \omega = \sqrt{k/m}.<br />
<br />
An important point to note here is that the above solutions do not represent a general solution for the displacement of the HO, rather they represent a family of solutions. The coefficient A is not constant throughout the whole motion, instead it is constant throughout each half oscillation.<br />
<br />
Before we process to assembling the piecewise smooth solution, can I get an agreement from <b>toastie</b> and <b>diazona</b> that we are all on the same page?
