Lazy square roots of power series (2007)

I can’t understand how this is used. Eg his expression for r_t in the inverse at the start. How would you use this in a program to get the inverse? Guess just pointing out that you can iteratively find one coefficient at a time.

Also the differential Eqn looks integrable.

You could do all that, or you could just collect like powers on one side and set the collected coefficients equal to zero, like a mathematician would.