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Math Help - Diffrence variable seperatable and non homogeneous sub catagory.

  1. #1
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    Diffrence variable seperatable and non homogeneous sub catagory.

    In my source, which I'm reading, the classification of differential equation; for the aid to solving it is given as Variable separable (V.S), non variable separable, homogeneous (there are more, but right now I'm concerned with this only)


    The solution given for homogeneous differential equation is 'convert the homogeneous differential equation into variable separable form'.


    So what's the difference between non variable separable and homogeneous?


    I see them as the same thing :-o




    I thought, that if its a homogeneous equation of degree 1 and order 1 (I'm currently solving only these sort of differential equations), then it can always be solved, but then, what's the other condition that exists? Equation of the form non variable separable, homogeneous are identical!...why do we have a sub category?


    Finally the procedures of solving non variable separable and homogeneous differential equations are the same.


    I see the same case with non homogeneous.
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  2. #2
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    For first order differential equations, an equation is said to be "separable" (what you are calling "variable separable) if it can be written in the form \frac{dy}{dx}= \frac{f(x)}{g(y)} or, equivalently g(y)dy= f(x)dx so that each side can be integrated separately. A first order differential equation, of the form [tex]\frac{dy}{dx}= f(x,y)[/quote] is said to be "homogeneous" if replacing x and y in f(x,y) by cx and cy respectively does not change the equation: that is, if the "c" cancels out.

    "Equation of the form non variable separable, homogeneous are identical!." is not true. For example, \frac{dy}{dx}= x+ y is a "non variabel separable equation" that is not homogenous. The important thing about homogeneous equations is that they can be changed to separable equations.

    \frac{dy}{dx}= \frac{x+y}{y-x} is homogeneous (since (cx+cy)/(cy- cx)= c(x+y)/c(y-x)= (x+y)/(y-x)) but not separable. What is true is that if we introduce the new dependent variable v= y/x, so that y= xv, dy/dx= x dv/dx+ v, then dividing both numerator and denominator by x gives (1+ y/x)/(y/x- 1)= (1+ v)/(v- 1) so the equation becomes x dv/dx+ v= (1+v)/(v-1) which IS separable: every homogeneous equation can be changed to a separable equation by that substitution.

    (Caution: Unfortunately, the word "homogeneous" is also used in a very different sense for differential equations of higher order.)
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  3. #3
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    You mean, usually, non variable separable equations are non homogeneous, but there can be non V.S and homogeneous, for whom procedures are a bit different (actually the procedures are identical).

    Also we have another sub category of solving non homogeneous, who's procedures are a lot different.

    So, the simple forms -- like V.S and non V.S....do not have a difference (with non homogeneous) since they are non homogeneous! The procedures are almost the same! Even if there were homogeneous, there procedures for solving them would have been the same!

    It should have been classified as 'another form' of non V.S.
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  4. #4
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    People pls!...I need this!
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