How to show an equation is homogeneous.

**shadow_2145** · Sep 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** · Sep 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** · Sep 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** · Sep 12th 2008, 05:05 AM

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

Thread: How to show an equation is homogeneous.

LinkBack

Thread Tools

Display

How to show an equation is homogeneous.

Solution for (2)

Similar Math Help Forum Discussions

show columns are linearly independent if homogeneous system has only trivial solution

using the homogeneous equation...

homogeneous equation

[SOLVED] Is x^3 + y^3 -xy^2 dy/dx = 0 a homogeneous equation?

non-homogeneous equation

Search Tags