Homogenous Equation to General Solution

For 10 points
Find the General Solution of this Equation:

(x*sin^2(y/x) + y)dx - xdy = 0

x * sin(y/x)^2 * dx + y * dx - x * dy = 0
x * sin(y/x)^2 * dx = x * dy - y * dx
(x/x^2) * sin(y/x)^2 * dx = (x * dy - y * dx) / x^2
dx / x = csc(y/x)^2 * (x * dy - y * dx) / x^2

y/x = t
(x * dy - y * dx) / x^2 = dt

dx / x = csc(t)^2 * dt

Now integrate

ln|x| = -cot(t) + C
ln|x| = -cot(y/x) + C
cot(y/x) = C - ln|x|
y/x = arccot(C - ln|x|)
y = x * arccot(C - ln|x|)

(x*sin^2(y/x) + y) dx = x dy

u = y/x

u + xu' = y'

(x*sin^2(u) + ux) = xu + x^2u'

x*sin^2(u) = x^2(du/dx)

1/x dx = csc^2(u) du

Integrating both sides:

ln|x| + C = -cot(u)

u = tan(-1/(ln|x| + C))

y = x*tan(-1/(ln|x| + C))