Which of the following statements is correct?

f(x,y) = x²*e^(x/y) + y³/x + y²*ln(y/x) is a homogeneous function of degree two.
f(x,y) = (sin y + x)/(sin 2y + x*cos y) is a homogeneous function of degree one.
x*sin(y/x) dy + (y*sin(y/x) - x) dx = 0 is a homogeneous differential equation.
f(x,y) = e^x + tan⁻¹(y/x) is a homogeneous function of degree zero.

Correct answer: x*sin(y/x) dy + (y*sin(y/x) - x) dx = 0 is a homogeneous differential equation.

Solution

Only the third option is genuinely homogeneous: both M = y*sin(y/x) - x and N = x*sin(y/x) are homogeneous of degree one, making the DE homogeneous.