First off mathematics isn't my field, so if I've made any rookie mistakes please forgive me! (also I'm not sure what tags to use)
I have a set of equations that boil down to this;
$s = \int Vdt$
$V = \int adt$
$a = \frac{c}{s} + d$
(where $c$ and $d$ are constants)
My problem is how do I get an equation for $s$ with respect to $t$ (or visa versa), which I thought was easy; just rearrange as so;
$V = (\frac{c}{s}+d)\times t$
$s = (\frac{c}{s}+d)\times t^2$
$s = \frac{c}{s}\times t^2 + d\times t^2$
$1 = \frac{c}{s^2}\times t^2 + \frac{d}{s}\times t^2$
$s^2 = c\times t^2+d\times s\times t^2$
$s^2 - d\times t^2\times s-c\times t^2 = 0$
The use the quadratic equation. However I've used the quadratic equation and rule out on of the results since it would give negative values for s (which isn't possible in my case) and I'm left with a single equation. When I put this equation into excel and try and plot $s$ wrt $t$ everything looks good until a certain time is reached then the equation makes no sense.
I expect to see a fast rise at the start and then for the change in s to decrease as the time passes. In reality after some time the whole system would plateau out and then stay that way irrespective of how much more time passes. However if I continue to increase the time then the s value starts decreasing and goes negative.
Any pointers on where I might have gone wrong; I'm doing this to understand the maths, so I've tried to keep the values as abstract as possible ($c$ and $d$ are a bunch of constants lumped together).
The differential equation that covers this is $$\frac{d^2s}{dt^2}=\frac cs+d$$ Multiply be $ds/dt$ and integrate, to get $$\frac12\left(\frac{ds}{dt}\right)^2=c*\log |s|+d*s+C_1\\ t=C_2+\int\frac{ds}{\sqrt{2c\log |s|+2d*s+2C_1}}$$ Unfortunately, I don't know how to do that integral.