How to show an equation is homogeneous.

**shadow_2145** · September 11th 2008, 09:58 PM

If the right side of the equation dy/dx = f(x,y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous.

(1) $\frac{dy}{dx} = \frac{x^2+xy+y^2}{x^2}$

$\frac{dy}{dx} = 1 + \frac{y}{x} + \frac{y^2}{x^2}$

That one was easy.

(2) $\frac{dy}{dx} = \frac{4y-3x}{2x-y}$

How do I show that this is homogeneous? I know there is a simple way, but I can't remember for the life of me. Thanks!

**Showcase_22** · September 12th 2008, 01:50 AM

I had a crack at it and got this far:

There's probably a better way of doing it than this (mainly because this didn't get an answer).

hope this helps......somehow.

**bkarpuz** · September 12th 2008, 03:11 AM

Originally Posted by shadow_2145

If the right side of the equation dy/dx = f(x,y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous.

(1) $\frac{dy}{dx} = \frac{x^2+xy+y^2}{x^2}$

$\frac{dy}{dx} = 1 + \frac{y}{x} + \frac{y^2}{x^2}$

That one was easy.

(2) $\frac{dy}{dx} = \frac{4y-3x}{2x-y}$

How do I show that this is homogeneous? I know there is a simple way, but I can't remember for the life of me. Thanks!

Please see the followings
$\frac{dy}{dx} = \frac{4y-3x}{2x-y}$
..... $=\frac{4y}{2x-y}-\frac{3x}{2x-y}$
..... $=\frac{4}{2\frac{x}{y}-1}-\frac{3}{2-\frac{y}{x}}$
..... $=\frac{4}{2\frac{1}{\frac{y}{x}}-1}-\frac{3}{2-\frac{y}{x}}$

**Krizalid** · September 12th 2008, 05:05 AM

For the second one, just divide by $x,$ top & bottom.

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