y'=z+xz'\\ z+xz'=\dfrac{x+2xz}{2x-xz}\\ xz'=\dfrac{1+z^2}{2-z}\\ \dfrac{z-2}{1+z^2}dz=-\dfrac{dx}{x}\\ (*) \int \dfrac{z-2}{1+z^2}dz=\dfrac{1}{2}\int \dfrac{2z}{1+z^2}dz-2\int \dfrac{1}{1+z^2}dz=\dfrac{1}{2}ln(1+z^2)-2arctg(z)+C_1\\ \dfrac{1}{2}ln(1+z^2)-2arctg(z)+lnx=C\\ \dfrac{1}{2}ln(1+\dfrac{y^2}{x^2})-2arctg(\dfrac{y}{x})+lnx=C" alt="y=xz=>y'=z+xz'\\ z+xz'=\dfrac{x+2xz}{2x-xz}\\ xz'=\dfrac{1+z^2}{2-z}\\ \dfrac{z-2}{1+z^2}dz=-\dfrac{dx}{x}\\ (*) \int \dfrac{z-2}{1+z^2}dz=\dfrac{1}{2}\int \dfrac{2z}{1+z^2}dz-2\int \dfrac{1}{1+z^2}dz=\dfrac{1}{2}ln(1+z^2)-2arctg(z)+C_1\\ \dfrac{1}{2}ln(1+z^2)-2arctg(z)+lnx=C\\ \dfrac{1}{2}ln(1+\dfrac{y^2}{x^2})-2arctg(\dfrac{y}{x})+lnx=C" align="absmiddle" class="latex-formula">