A quick algebra question derived from a solutions by substitution, differential equation

Question

The question I have starts with the problem $$x\frac{dy}{dx}=y+\sqrt{x^2-y^2},\space x>0$$ After substituting in the values for $y$ and $dy$ and multiplying out $$xu+x^2\frac{du}{dx}=ux+\sqrt{x^2-u^2x^2}$$Then factoring both the $x$ and $u$ terms $$x(u+x)\frac{du}{dx}=ux+x\sqrt{1-u^2}.$$ so my question is... $$x(u+x)\frac{du}{dx}=x(u+1)\sqrt{1-u^2}$$

Does this still work and if it does have I only complicated it further.

Answer 1

$$xu+x^2\frac{du}{dx}=ux+\sqrt{x^2-u^2x^2}$$Then factoring both the $x$ and $u$ terms $$x(u+x)\frac{du}{dx}=ux+x\sqrt{1-u^2}$$

It should be $$x\left(u+x\frac{du}{dx}\right)=ux+x\sqrt{1-u^2},$$ which reduces to $$x\frac{du}{dx}=\sqrt{1-u^2}.$$ Can you complete now?

Answer 2

You should use that $xu=ux$ and simplify from the second formula to $$ x\frac{du}{dx}=\sqrt{1-u^2} $$ which is separable. Thus you avoid the strange error in the last formula completely.