Solving the Brachistochrone problem with friction

[physics_cosmos]

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence.

I am asking for help understanding how the solution to the differential equation obtained from applying the Euler-Lagrange equation to the integrand of the the integral representing the total time of descent is obtained. This differential equation can be found on step (30) of the page. I am asking for help in understanding the next step, how setting dy / dx = cot(theta/2) results in the given parametric forms for x and y (in terms of theta), as given in steps (32) and (33).

 

[jedishrfu]

Is that step simply choosing an integrating factor that fits the differential equation you're trying to solve?

 

[physics_cosmos]

Is that step simply choosing an integrating factor that fits the differential equation you're trying to solve?

I don't believe the differential equation is linear, or resolvable into linear form, so I don't think that that would help me get to a solution. I hope you can help me find another way, though.