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Math Help - Concept of Homogeneity

  1. #1
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    Concept of Homogeneity

    I'm reading through this proof that any homogeneous equation y' = f(x,y) can be rewritten as y' = g(y/x). It says that, by homogeneity, f(x,y) = f(tx, ty) for all real t (I think it should have the proviso that t \ne 0, but it doesn't.). It then claims--and here's the part I don't get--that an appropriate choice of t may be 1/x. I thought t was some real number, not a real-valued function. Am I missing something important here, or should I just take homogeneity to mean something like f(x, y) = f(tx, ty) for any function t(x,y)?
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    Nuthin?
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    I think you should have t^{\alpha}f(x,y) \ \alpha\in\mathbb{R}
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    The book starts with what I gave but then amends it later in a way similar to what you have. So no clue about exactly what homogeneity means?
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  5. #5
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    This homogeneous refers to if both coefficient functions M and N are of the same degree.

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

    Homogeneous means the same; hence, the degrees have to be the same. However, this isn't the same as a homogeneous equation though.
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