dy/dx = (x^2+3y^2)/(2xy)
dy/dx = (1+3*(y^2/x^2))/(2(y/x))
Consider u = (y/x)
In this way, dy/dx = x(du/dx) +u = (1+3*(y^2/x^2))/(2(y/x)) = (1+3u^2)/2u
so, x(du/dx) = (1+3u^2)/2u) - u
x(du/dx) = (1+u^2)/2u
\int((2u/(1+u^2))du = \int(1/x)dx
ln|1+u^2| = ln|x| + c
When u = (y/x),
ln|(x^2+y^2)/(x^2))| = ln|x|+c
ln|x^2 + y^2| - 2ln|x| = ln|x| + c
ln|x^2 + y^2| - 3ln|x| = c
ln|x^2 + y^2| = 3ln|x| + c
e^(ln(x^2 + y^2)) = e^(3ln|x|) +e^c
x^2 + y^2 = x^3 + c
