I have solved a differential equation recently, which left me with this whopper of inverse function to figure out. I know what $c$ is, I just haven't calculated its exact value based on the initial conditions yet.
(1): $-\sqrt{y} \sqrt{5615673-y}+5615673\cdot\tan^{-1}{\left(\frac{\sqrt{y}}{\sqrt{5615673-y}}\right)}=11133.58 t+c$
This equation looked pretty hopeless at first, but then I saw the connection between this and forms like $\sin x = u$, $\cos x = \sqrt{1-u^2}$
Using that connection I looked for a transformation until I found:
(2): $\sqrt{y} = \sqrt{5615673}\sin\alpha$
Which makes:
(3): $\sqrt{5615673 - y} = \sqrt{5615673}\cos\alpha$
Making those substitutions the equation becomes:
(4): $-5615673\cos\alpha\sin\alpha+56165673\alpha = 11133.58 t + c$
From there, it was simple to get to:
(5): $\sin\beta - \beta = s$
Obviously there's no inverse for $\sin x - x$, but is there another approach to solving the 1st equation?
If not, is there a good technique for approximating this sort of thing, or should I just go back to the differential equation and approximate that?