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Thread: Linear and homogeneous DE

  1. #1
    Mar 2010

    Question Linear and homogeneous DE

    Why is there a difference? I didn't find any explanation on the web.

    So Linear homogeneous DE are different then just homogeneous DE. Why are we using the same word, and why are they called like that?

    Thanks for the explanation. (Why are exact DE called like that?)

    I really can't remember these names because I don't understand where they come from.
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  2. #2
    MHF Contributor

    Apr 2005

    Re: Linear and homogeneous DE

    I think you are referring to the uses of the word "homogeneous" in two different ways:

    1) A first order differential equation, of the form dy/dx= f(x, y) is said to be homogeneous if f(x, y) can be expressed as a function of the single variable y/x.

    2) A linear equation, of any order, is of the form $\displaystyle a_n(x) d^ny/dx^n+ a_{n-1}(x)d^{n-1}y/dx^{n-1}+ \cdot\cdot\cdot+ a_1(x) dy/dx+ a_0(x)y= f(x)$ and is homogeneous if f(x) is identically 0.

    The reason for the two uses is historical. Specifically, it is due to the fact that the word "homogeneous" is itself used in mathematics in several different ways, all meaning "the same" in some way.

    The idea of an "exact" equation comes from the idea of an "exact" differential. If f(x, y) is a function of the two variables, x and y, but x and y are themselves functions of the single variable t, f(x(t), y(t)), then the derivative with respect to t is given by the "chain rule", $\displaystyle \frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}$. The "differential" can be written $\displaystyle df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy$. We can write any first order differential equation of the form p(x, y)dx+ q(x, y)dy= 0 which looks an awful lot like that differential but may not be. Such a differential equation is called "exact" if there is a function, f(x, y) such that $\displaystyle df= p(x, y)dx+ q(x, y)dy$.
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