I'm trying to solve differential equation to get EOM for a dynamical system. Firstly, I reduced order for my equation by substitution $$ \dot{v}_x(t) = \frac{\mathrm{d}x}{\mathrm{d}t}. $$
I solved it and got this equation: $$ \frac{\mathrm{sgn}(v_x)}{v_x} = At + B, $$ where $A,B$ are const.
I know, that I can now look at two cases with $v_x>0$ or $v_x<0$, but I wonder if I can calculate it with signum function. I habe no idea what to do, when I come back and put $$ v_x = \frac{\mathrm{d}x}{\mathrm{d}t} $$
How to calculate integral of derivative inside signum function?
$$\frac{\text{sgn} (v_x)}{v_x} $$ is always positive, so $At+B\gt 0$. There are two solutions to the equation you got, owing to the fact that $v_x \mapsto -v_x$ doesn’t change the equation:
$$v_x = \pm \frac{1}{At+B} =\frac{dx}{dt} \\ x(t) = \pm \frac 1A \ln(At+B)+C $$