Question

My textbook says
M(x,y)dx+N(x,y)dy=0
is homogenous if
M(tx,ty)=tαM(x,y)
and
N(tx,ty)=tαN(x,y)
. What does that mean?
I know that if the function equals
F=x/y
then it's homogenous. How do those two rules relate?

Answer 1

homogenous .... cant recall what it means; but its got the property that it can be scaled

Answer 2

x+y^2 is not homogenous because when we include the "t" we get:

tx + t^2y^2

t(x+ty) is not the same

Answer 3

Are you talking about differential equations?

Answer 4

no; just a property of homogenoues and nonhomogenous equations; i cant recall how that relates into differential equations tho

Answer 5

When you wrote tx+t^2y^2, should that be tx+ty^2?

Answer 6

almost: prolly more proper like this:

tx + (ty)^2

Answer 7

the t helps to visuallize the homogenous property. If we can pull out the "t" and the equation remains the same; it is considered to be homogoneus

Answer 8

x/y is homogen.. because: tx/ty = x/y ... so the equation remains unchanged when we test for t

Answer 9

So in relation to the question:

the diff equation is homogen if M and N are homogen

Answer 10

My book says that x^3 +y^3 +1 isn't homogenous. Is that because if you put a "t" by the x and y, you can't take it out by saying t^3(x,y) becasue then the 1 will be a t^3, which isn't true?

Answer 11

the +1 prolly messes it up; yeah

Answer 12

I think it's easier (for me, at any rate) just to say that if when you put (whatever) in derivative form, there is some way of writing it as a function y/x.

Answer 13

It definnitely is but my professor said we might have to show the other way on a exam :-)

Answer 14

he forgot the "n" so it doesn't count lol anyways thanks

Answer 15

Heh, if you delete it, it didn't happen....

Answer 16

:) what didnt happen

Answer 17

I guess it's the same thing really, the t has to depend on y/x anyway.
f(x,y) = f(1/x x, 1/x y) = f(1,y/x) -> f(x,y) = h(y/x).