The equation

$y^{\text{'}}=\frac{6x^2+3y^2}{xy}$

can $be$ written $in$ the form $y^{\text{'}}=f(\frac{y}{x}),i.e.,itis$ homogeneous, $sowe$ can use the substitution $u=\frac{y}{x}to$ obtain a separable equation with dependent variable

$u=u(x).$

Introducing this substitution and using the fact that $y^{\text{'}}=x{u}^{\text{'}}+uwe$ can write $(*)as$

$y^{\text{'}}=x{u}^{\text{'}}+u=f(u)$ where $f(u)=$

Separating variables $we$ can write the equation $in$ the form

$g(u)du=\frac{dx}{x}$