You of all people should know that this is a special case of Kamke's differential equation 6.73! But indeed, when you wrote the book, there wasn't much known about this equation.
Here's what is said in the book, translated in English:
..
[itex]u^{''}=y^n x^{1-n}[/itex]
for the special case that n=2, one solution is 2/x^2
Well, I am glad that after 65 years, you've managed to correct your mistake.
By the way, also Maple gives a very complicated implicit solution, so I can't help you further.