\[\begin{array}{l}y = {z^2}\\2z' = \frac{x}{z} + \frac{{{x^3}}}{{{z^3}}} \Leftrightarrow z' = \frac{1}{{2 \cdot \frac{z}{x}}} + \frac{1}{{{{\left( {\frac{z}{x}} \right)}^3}}}{\text{ - однородное}}{\text{,}}\\{\text{замена }}\frac{z}{x} = u \Leftrightarrow z = ux\\u'x + u = \frac{1}{{2u}} + \frac{1}{{{u^3}}} \Leftrightarrow u'x = \frac{{{u^2} - 2{u^4} + 2}}{{2{u^3}}} \Leftrightarrow \frac{{2{u^3}du}}{{{u^2} - 2{u^4} + 2}} = \frac{{dx}}{x} \Leftrightarrow \\\int {\frac{{2{u^3}du}}{{{u^2} - 2{u^4} + 2}}} = \ln Cx\end{array}\]