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Show that this equation is homogeneous?

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2 Answers

dy/dx=(4y - 3x)/(2x - y)

(2x - y)dx - (4y - 3x)dy = 0

If you multiply x and y by a scalar, say A, you get

(2Ax - Ay)dx - (4Ay - 3Ax)dy = A(2x - y)dx - A(4y - 3x)dy

= A[(2x - y)dx - (4y - 3x)dy] = 0

Since this scales the whole left hand side by A, the equation is homogeneous.

In general:

A polynomial P(v) with all terms having degree n is homogeneous as P(Av) = AnP(v) for all vectors v and scalars A.

A differential equation of the form dy/dx = M(x,y) / N(x,y) is homogeneous if M and N are homogeneous of the same degree.

Proof:

Let M and N be both homogeneous polynomials of degree K.

Rewrite M(x,y) dx - N(x,y) dy = 0

Now M(Ax,Ay) dx - N(Ax,Ay) dy = AKM(x,y) dx - AKN(x,y)dy = AK[M(x,y) dx - N(x,y) dy].