Solving differential equations with signum Is there a general solution to:
$A x''+ B x' + C \mathop{\rm sgn}(x')+ D x=0$  where $\mathop{\rm sgn}(x')$ is the sign of $x'$
 A: The solution will satisfy $A x'' + B x' + C + D x = 0$ in intervals where $x' > 0$ and $A x'' + B x' - C + D x = 0$ in intervals where $x' < 0$.  If $B^2 - 4 A D > 0$, for example (the overdamped case), the solution of $A x'' + B x' + C + D x = 0$ are of the form $x = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t} - C/D$ for arbitrary constants $c_1$ and $c_2$, where $\lambda_i$ are the roots of the quadratic $A \lambda^2 + B \lambda + D$.  Then $x' = c_1 \lambda_1 e^{\lambda_1 t} + c_2 \lambda_2 e^{\lambda_2 t} = 0\ $ for
$t = \ln(-c_1 \lambda_1/(c_2 \lambda_2))/(\lambda_2 - \lambda_1)\ $ in the case where
$c_1 \lambda_1/(c_2 \lambda_2) < 0$, or never if $c_1 \lambda_1/(c_2 \lambda_2) > 0$.  Thus we have solutions which are either monotonic solutions of $A x'' + B x' + C + D x = 0$ or of $A x'' + B x' - C + D x = 0$, or piecewise solutions (a solution of one of these equations on one interval and a solution of the other equation on its complement, joined at a point where $x' = 0$ (on both sides).
In the underdamped case, things can be more complicated because we will have infinitely many intervals.
A: The ODE you have presented is called a piecewise affine (PWA) dynamical system in the control theory community. In particular, since you used the signum function, what you have is a relay feedback system. These models are quite common in electrical engineering (hence the word "relay"), but they seem to be immune to analysis, a cemetery of theories, so to speak. 
