Final answer:
A linear differential equation is homogeneous if the coefficient of y is zero, a separable differential equation is homogeneous if the functions of x and y are equal, and an exact differential equation is homogeneous if the functions are of x/y or y/x.
Step-by-step explanation:
A linear differential equation of the form dy/dx + P(x)y = Q(x) is homogeneous if Q(x) = 0 for all x. In other words, the coefficient of y in the equation is zero. For example, the equation dy/dx + 2xy = 0 is homogeneous because the coefficient of y is 0.
A separable differential equation can be written in the form dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively. The equation is homogeneous if f(x) = g(y). For example, the equation dy/dx = x/y is separable and homogeneous because x/y = y/y = 1.
An exact differential equation can be written in the form M(x,y)dx + N(x,y)dy = 0. The equation is exact if and only if the partial derivatives of M(x,y) and N(x,y) with respect to y and x, respectively, are equal. This means that the equation is homogeneous if M(x,y) and N(x,y) are functions of x/y or y/x. For example, the equation ydx + xdy = 0 is exact and homogeneous because y/x = 1.