For (14), we are given:

$$(x^2+1)yU_{x}+(y^2+1)xU_{y}=0$$

If I didn't make a mistake, the characteristic equation is:

$$ C=\frac{1+x^2}{1+y^2}$$

But now I'm a little confused how to solve:

$$ du=\frac{dx}{y(1+x^2)}$$ Wrong. And be consistent: either $u$ or $U$ V.I.

Anyone have an idea? If we solve y in terms of x we get two values because of the root, so how should we proceed?