Originally Posted by
Ackbeet I'm not sure I quite agree. I get
$\displaystyle \displaystyle{\frac{d}{dx}\left(\frac{y^{2}}{x^{2} }\right)=\frac{2x^{2}yy'-2xy^{2}}{x^{4}}=\frac{2xyy'-2y^{2}}{x^{3}}=\frac{2}{x}\left(\frac{xyy'-y^{2}}{x^{2}}\right)=\frac{2}{x}\sqrt{4+\left(\fra c{y}{x}\right)^{2}}.}$
That last equality comes from the DE, from arranging the LHS to look like what's in the parentheses. At this point, you could make the substitution $\displaystyle u(x)=y(x)/x.$
I was thinking quotient rule for derivatives, but was getting hung up on not having the x squared term in the denominator, which you have to have in order to make the coefficients the same in the numerator.