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Thread: Separable DE

  1. April 2nd 2010, 03:37 PM #1
    cdlegendary
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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. April 2nd 2010, 03:47 PM #2
    Anonymous1
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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. April 2nd 2010, 04:08 PM #3
    Krizalid
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    that's wrong, the ODE actually writes as \frac{dy}{y-y^{3}}=dx,
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  4. April 2nd 2010, 04:16 PM #4
    Anonymous1
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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. April 2nd 2010, 04:45 PM #5
    cdlegendary
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    uhh yeah there aren't any initial conditions
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  6. April 2nd 2010, 04:48 PM #6
    skeeter
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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. April 2nd 2010, 05:23 PM #7
    cdlegendary
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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. April 2nd 2010, 05:26 PM #8
    cdlegendary
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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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