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Math Help - Conserved quantity

  1. #1
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    Conserved quantity

    I have read a paper recently in which the authors consider a system of differential equations show below.


    <br />
x'=x+a-y
    <br />
y'=x(b-2y)/c<br /> <br />
    Where

    a,b,c are constants.

    They claim that the quantity:


    <br />
y^2/2-y(a+x)^2+bx^2/2



    is conserved. After trying to derive this quanity from the governing equations, I can't work out where the 'c' disappears. Any thoughts would much much appreciated.
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  2. #2
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    Very sorry, I've posted the question wrong. My problems remain though...

    The actual question should have read:

    x'=x^2+a-y
    y'=b(x-2y)/c

    The conserved quantity is:

    <br />
bx^2/2-y(a+x^2)+y^2/2<br />

    Here's what I did with it though, maybe this will help...
    If you ignore the 'c'.

    Then you can do the following.

    x'=x^2+a-y

    so
     <br /> <br />
x'-x^2-a+y=0<br />

    then multiply through by [tex] y' [\math] to get

    x'y'-x^2y'-ay'+yy'=0

    then substitute (ignoring c)  y'=x(b-2y) to get

     <br />
bxx'-2xx'y-x^2y'-ay'+yy'=0<br />

    Which you can integrate to get:

    <br />
bx^2/2-ay-x^2y+y^2/2=constant<br />

    Which gives the result. Don't know how to do it with the c though.
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  3. #3
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    Quote Originally Posted by kyz1024 View Post
    Very sorry, I've posted the question wrong. My problems remain though...

    The actual question should have read:

    x'=x^2+a-y
    y'=b(x-2y)/c (*)

    The conserved quantity is:

    <br />
bx^2/2-y(a+x^2)+y^2/2<br />

    Here's what I did with it though, maybe this will help...
    If you ignore the 'c'.

    Then you can do the following.

    x'=x^2+a-y

    so
     <br /> <br />
x'-x^2-a+y=0<br />

    then multiply through by [tex] y' [\math] to get

    x'y'-x^2y'-ay'+yy'=0

    then substitute (ignoring c)  y'=x(b-2y) (*) to get

     <br />
bxx'-2xx'y-x^2y'-ay'+yy'=0<br />

    Which you can integrate to get:

    <br />
bx^2/2-ay-x^2y+y^2/2=constant<br />

    Which gives the result. Don't know how to do it with the c though.
    I see a conflict. See the above in red (*)
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