Simplifying solutions I am given the differential equation $$\frac{dz}{dx} = m(c_{1}-z)(c_{2}-z)^{\frac{1}{2}}, z(0) =0$$ and have arrived at a solution: $$z(x) = c_{2} - (c_{1}-c_{2})\tan^{2}{\left[\arctan{\left(\frac{\sqrt{c_{2}}}{\sqrt{c_{1}-c_{2}}}\right)} - \frac{mx}{2}\sqrt{c_{1}-c_{2}}\right]}.$$ I was wondering if there was any more simplification that could occur here since we have the $\arctan$ within the $\tan$. I have tried using the identity for $\tan{(A-B)}$ but this doesn't seem to simplify matters.

If I may, I would have integrated the differential equation ignoring first the boundary condition; this leads to 
$$z(x)=\text{c2}-(\text{c1}-\text{c2}) \tan ^2\left(\frac{1}{2} \sqrt{\text{c1}-\text{c2}} (K-m
   x)\right)$$ for which the boundary condition gives $$(\text{c1}-\text{c2}) \tan ^2\left(\frac{1}{2} \sqrt{\text{c1}-\text{c2}}
   K\right)=\text{c2}$$ from which $K$ can be easily extracted to give your expression  
I must confess that I wrote this thinking about the manner I should code it for numerical aplications. It is just a personal opinion but if find it easier to read.
