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Math Help - Separable DE

  1. #1
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    Separable DE



    For the equation above, I need to find all the constant solutions. How do I go about doing this? I've tried putting all similar terms on one side and then integrating, but it doesn't seem to work out nicely. Any help is greatly appreciated, thanks in advance!
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  2. #2
    Super Member Anonymous1's Avatar
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    \int y' = \int y-y^3

    \Rightarrow y=  \frac{2y^2 - y^4}{4} + C

    \Rightarrow \frac{y}{4}(y^3-2y+4)=C

    \Rightarrow \frac{y}{4}(y^3-2y+4)-C=0

    Code:
    syms y c
    solve('(y/4)*(y^3-2*y+4)-c=0',y)
     
    ans =
     
     - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2)
       1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2) - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2)
       1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2)
       1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2)
    umm... lol. Your sure your not given any initial conditions, y(x)=0?
    Last edited by Anonymous1; April 2nd 2010 at 04:02 PM.
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  3. #3
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    that's wrong, the ODE actually writes as \frac{dy}{y-y^{3}}=dx,
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  4. #4
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by Krizalid View Post
    that's wrong, the ODE actually writes as \frac{dy}{y-y^{3}}=dx,
    Figured it had to be . So your pretty good at calc, huh?
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  5. #5
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    uhh yeah there aren't any initial conditions
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  6. #6
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    Quote Originally Posted by cdlegendary View Post


    For the equation above, I need to find all the constant solutions. How do I go about doing this? I've tried putting all similar terms on one side and then integrating, but it doesn't seem to work out nicely. Any help is greatly appreciated, thanks in advance!
    if y = C , then y' = 0 ... correct?
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  7. #7
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    Quote Originally Posted by skeeter View Post
    if y = C , then y' = 0 ... correct?
    ah yes. I think I know where you're going with that
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  8. #8
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    so the solutions are just -1, 0, 1. wow haha. I was definitely thinking too hard about that. thanks!
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