Understanding Homogeneous Differential Equations

[magnifik]

i am having trouble trying to classify whether or not a differential equation is homogeneous.

i know that if it is, f(tx, ty) = tⁿf(x, y)
but i don't really know what this means

for example,
why is f(x, y) = e^xy not homogeneous
but f(x, y) = e^x/y is homogeneous??

does that mean a function like
f(x, y) = e^x/y dx + 3 dy would be homogeneous?

 

[vela]

i am having trouble trying to classify whether or not a differential equation is homogeneous.

i know that if it is, f(tx, ty) = tⁿf(x, y)
but i don't really know what this means

for example,
why is f(x, y) = e^xy not homogeneous

Because f(tx, ty) = e^(tx)(ty) = e^t2xy ≠ tⁿ e^xy = tⁿf(x,y) for any value of n.

but f(x, y) = e^x/y is homogeneous??

Because in this case f(tx, ty) = e^(tx)/(ty) = e^x/y = t⁰f(x,y)

does that mean a function like
f(x, y) = e^x/y dx + 3 dy would be homogeneous?

That doesn't really make sense.

 

[magnifik]

how does it not make sense? :\ that's an example in my textbook for a homogeneous differential equation

 

[fzero]

how does it not make sense? :\ that's an example in my textbook for a homogeneous differential equation

The term "homogeneous" is used in a slightly different way in the case of linear differential equations. A linear differential equation for a function y(x) is called homogeneous if it is invariant under the rescaling transformation y(x) -> t y(x). This means that an equation

a_n (x) \frac{d^ny(x)}{dx^n} + \cdots + a_1(x) \frac{dy(x)}{dx} +a_0(x) y(x)= f(x)

is homogeneous only for f(x)=0.

The usage isn't completely different from the one for functions, since homogeneity still refers to a behavior under a rescaling. But for ODEs it does not involve a rescaling of the coordinate x.

 

[vela]

how does it not make sense? :\ that's an example in my textbook for a homogeneous differential equation

I'd be surprised if it was written in your textbook exactly as you wrote it here.

Here are some web pages that explain homogeneous differential equations:

http://www.cliffsnotes.com/study_guide/First-Order-Homogeneous-Equations.topicArticleId-19736,articleId-19713.html
http://www.tutorvista.com/math/solving-homogeneous-differential-equations

What they say is the differential equation

M(x,y) dx + N(x,y) dy = 0

is homogeneous if M(x,y) and N(x,y) are homogeneous functions of the same degree. So in the differential equation

e^x/y dx + 3 dy = 0 [not f(x,y)]

you have M(x,y) = e^x/y and N(x,y) = 3. Both functions are homogeneous functions of degree 0, so the differential equation is a homogeneous differential equation.

You can transform such an equation into a separable one by using the substitution y=xv.


Note that fzero's post explains the usual meaning of "homogeneous differential equation." I'm assuming you're referring to the less common meaning based on your question about homogeneous functions of degree n.

 

[magnifik]

can a homogeneous differential equation be exact?

 

[fzero]

Sure, the functions M(x,y) and N(x,y) from vela's post must still be homogeneous functions of the proper degree and must also satisfy

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} .

The potential function will be homogeneous of one degree higher.