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'
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'
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.
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.