I'm curious about solving a differential equation for the displacement vs. time on the vertical movement of a slinky when it is suspended from rest and starts to oscillate.
I have gotten to these steps, but how do you solve the whole differential equation?
$mg$ - $cv^2$ - $kx$ = m$\frac{dv}{dt}$ , where m = mass of slinky, c = coefficient of air resistance, v = velocity, k = Hooke's coefficient of slinky, x = displacement
$g$ - $\frac{c}{m}v^2$-$\frac{k}{m}x$ = $\frac{dv}{dt}$
$1$ - $\frac{c}{mg}v^2$-$\frac{k}{mg}x$ = $\frac{dv}{gdt}$
Letting A = $\frac{c}{mg}$ and B = $\frac{k}{mg}$: $$\int \frac{1}{1-Av^2-Bx} \, dv$$=$$\int g dt$$= $gt$ + C, where C is the cosntant of integration.
Because the motion is periodic, $x$=$acoswt$ how could this be incorporated?