The DE that was studied three centuries ago by the Italian mathematician 'Conte' Jacopo Riccati has the form...

(1)

... and in general can be solved only if we know a particular solution of it. In that case if

is a particular solution of (1) and we set...

(2)

... after some algebraic steps we obtain from (1)...

(3)

... that is linear in z. In our case the DE is...

(4)

... and it is easy to verify that

is a solution. At this point if You set in (2)

and

You obtain the substitution...

(5)

... exactly as in your book...

Kind regards