Show that an equation is homogeneous?

Question

Show that (x^2+3xy+y^2)dx-x^2*dy=0 is homogeneous.

Answer 1

Looking at (x² + 3xy + y²)dx - x²dy = 0, note that is is already of the M(x,y)dx + N(x,y)dy.

Let v = <x,y>.

For the two functions M(v) = x² + 3xy + y² and N(v) = -x², we can establish homogeneity as follows.

M(av) = (ax)² + 3(ax)(ay) + (ay)² = a²x² + 3a²xy + a²y² = a²(x² + 3xy + y²) = a²M(v).

N(av) = -(ax)² = -a²x² = a²N(v).

Since M and N are the same degree, this same scalar multiplication property holds for the whole equation.

Thus M and N are homogeneous and so is your equation.