The problem is find the particular solution of $\frac{dy}{dx}$=$\frac{y-x}{y+x}$ when f(7)=7
For $\frac{dy}{dx}$=$\frac{y-x}{y+x}$, first i substitute y and $\frac{dy}{dx}$ to $ux$ and $x\frac{du}{dx}+u$ so the equation change to
$x\frac{du}{dx}+u$=$\frac{u-1}{u+1}$ now i think this equation is separable. $(u+1)du$=$\frac{-2}{x}dx$ Then I integration the equation and solve the problem but I wrong I think i solve the problem right way and I can't find my mistake what is the wrong part of my solution?
The mistake is that $$ \frac{u-1}{u+1}-u=-\frac{1+u^2}{1+u}. $$ So the new equation is $$ \frac{1+u}{1+u^2}\,u'=-\frac1x. $$ So you get $$ \arctan u + \frac12\,\log (1+u^2)=-\log x + c, $$ which gives you $$ \arctan \frac yx+\frac12\,\log(1+\frac{y^2}{x^2})=-\log x + c. $$ No hope of solving this explicitly. Since $y(7)=7$, $$c=\arctan 1+\frac 12\log2+\log7=\frac\pi4+\log(7\sqrt2).$$