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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?

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$$\frac{\text{sgn} (v_x)}{v_x} $$ 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:

$$v_x = \pm \frac{1}{At+B} =\frac{dx}{dt} \\ x(t) = \pm \frac 1A \ln(At+B)+C $$

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  • $\begingroup$ Thank you! Sometimes, the simplest solutions are the hardest to see! $\endgroup$
    – user718728
    Commented Mar 4, 2021 at 14:46
  • $\begingroup$ @mr_imp Sometimes all it takes is a little pondering. $\endgroup$
    – Vishu
    Commented Mar 4, 2021 at 15:08

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