Recently, I found that Google Copilot can solve a nonlinear but solvable equation like coulomb damping[1], just by searching for “r’‘(t)=-r(t)-sgn(r’(t))/10, r(0)=0, r’(0)=1” [2], whether Copilot’s time-dependent answer is correct or not aside.
Is it possible to refine the following command
DSolve[{r''[t] == -r[t] - Sign[r'[t]]/10, r[0] == 0, r'[0] == 1}, r[t], t]
to obtain the same analytic result in Mathematica v.14.0 ?
[1] https://en.wikipedia.org/wiki/Coulomb\_damping
[2] Google Search
This equation is problematic to my eye. The first alarm is that it contains Sign, which is discontinuous. Next, both Google and NDSolve claim that the solution becomes a nonzero constant after a certain instant, but this makes the two sides of the differential equation not the same:
eq = r''[t] == -r[t] - Sign[r'[t]]/10;
solNum = NDSolveValue[{eq, r[0] == 0, r'[0] == 1}, r, {t, 0, 20}]
Plot[solNum[t], {t, 0, 20}]
Plot[Evaluate[List @@ eq /. r -> solNum], {t, 15, 20}]
This is out of my expertise.
Thank you all for your comments. Google is giving the same answer as Professor Rogers again today. I hope that Mathematica’s DSolve will eventually be able to provide the correct answer.