Final answer:
The equation (x² + y²)dx + 2xy dy =0 is a homogeneous differential equation because all terms are of the same degree, which is two in this case.
Step-by-step explanation:
The equation provided, (x² + y²)dx + 2xy dy =0, needs to be analyzed to determine if it is separable or homogeneous. A separable equation can be written in the form f(y)dy = g(x)dx, where the variables can be separated on different sides of the equals sign. A homogeneous equation has the same degree for all its terms and can typically be solved by substitution.
In this case, after trying to separate the variables, we observe that the variables cannot be completely separated as both dx and dy terms are modified by expressions involving both x and y.
Checking if it's a homogeneous equation, we see that all the terms in the parentheses are of degree two (x² and y²), and the term 2xy is also of degree two since both x and y are to the first power. Given that all the terms are of the same degree, we can confirm that this is a homogeneous differential equation.