Looks like you're given
x cos(y/x) (y dx + x dy) = y sin(y/x) (x dy - y dx)
Like with your other equation, multiply by 1/x² on both sides:
cos(y/x) (y/x dx + dy) = y/x sin(y/x) (dy - y/x dx)
Then substitute z = y/x again, so y = xz and dy = x dz + z dx. You end up with a separable equation.
cos(z) (z dx + x dz + z dx) = z sin(z) (x dz + z dx - z dx)
cos(z) (2z dx + x dz) = xz sin(z) dz
2z cos(z) dx = x (z sin(z) - cos(z)) dz
(z sin(z) - cos(z)) / (z cos(z)) dz = 2/x dx
(tan(z) - 1/z) dz = 2/x dx
Integrate both sides:
∫(tan(z) - 1/z) dz = ∫ 2/x dx
-ln|cos(z)| - ln|z| = 2 ln|x| + C
ln|cos(z)| + ln|z| = -2 ln|x| + C
ln|z cos(z)| = ln(1/x²) + C
z cos(z) = C/x²
Solve in terms of y :
(y/x) cos(y/x) = C/x²
y cos(y/x) = C/x