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Math Help - Nonlinear nonexact first order ODE

  1. #1
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    Nonlinear nonexact first order ODE

    Hey
    How can I solve
    dy/dx = (3x^2 * y + y^2) / (2x^3 + 3xy)
    Hopefully you can understand that. It is clearly nonlinear. It is also nonexact and I cannot find an integrating factor that works. I also tried y=vx to no avail. Can someone help me out with this and point me in the right direction?
    Thanks
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Nonlinear nonexact first order ODE

    Find g(x,y) to make diff eq exact:

    x\left(2x^2+3 y\right)dx-y\left(3x^2+y\right)dy=0

    x\left(2x^2+3 y\right)\frac{1}{x y\left(3x^2+y\right)+2y x\left(2x^2+3 y\right)}dx-y\left(3x^2+y\right)\frac{1}{x y\left(3x^2+y\right)+2y x\left(2x^2+3 y\right)}dy=0

    Then solution is:

    x^3 y+ x y^2=c

    y\text{=}\frac{-x^3\pm \sqrt{4 c x+x^6}}{2 x}

    for c=1,2,3,....16





    also check out:

    f(x,y)=C+\frac{1}{7} \left(\log \left(x^2+y\right)+\log (x)\right)+\frac{2 \log (y)}{7}
    Attached Thumbnails Attached Thumbnails Nonlinear nonexact first order ODE-ode-nonlinear-nonexact.png   Nonlinear nonexact first order ODE-ode-nonlinear-nonexact2.png  
    Last edited by MaxJasper; October 9th 2012 at 06:11 PM.
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    Re: Nonlinear nonexact first order ODE

    How is that supposed to help? I am not familiar with diagrams like these for differential equations at all
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    Re: Nonlinear nonexact first order ODE

    Hi MaxJasper !
    There is a mistake in your first line (permutation of dx and dy).

    Hi Shanter !
    Your differential equation is solvable, but very arduously (too difficult to an home work).
    I suspect that there is a "-" missing in the wording :
    If the equation was : dy/dx = - (3x^2 * y + y^2) / (2x^3 + 3xy) then the integrating factor would be easy to find and the result : (x^3)(y^2)+(y^3)x = C
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Nonlinear nonexact first order ODE

    For ODE:
    dy/dx = (3x^2 * y + y^2) / (2x^3 + 3x y^2)



    Attached Thumbnails Attached Thumbnails Nonlinear nonexact first order ODE-ode-nonlinear-nonexact3.png   Nonlinear nonexact first order ODE-ode-nonlinear-nonexact4.png  
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    Re: Nonlinear nonexact first order ODE

    and I missed the negative, which really simplifies everything now that I can see that
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  7. #7
    Senior Member MaxJasper's Avatar
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    Question Re: Nonlinear nonexact first order ODE

    Quote Originally Posted by JJacquelin View Post
    Hi Shanter !
    Your differential equation is solvable, but very arduously (too difficult to an home work).
    I suspect that there is a "-" missing in the wording :
    If the equation was : dy/dx = - (3x^2 * y + y^2) / (2x^3 + 3xy) then the integrating factor would be easy to find and the result : (x^3)(y^2)+(y^3)x = C
    Hi folks,

    How did you solve the modified version with (-) sign included?

    Your solution: (x^3)(y^2)+(y^3)x = C
    results in :

    y'=-(3 x^2 y^2 + y^3)/(2 x^3 y + 3 x y^2)
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  8. #8
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    Re: Nonlinear nonexact first order ODE

    Quote Originally Posted by MaxJasper View Post
    How did you solve the modified version with (-) sign included?
    Your solution: (x^3)(y^2)+(y^3)x = C
    results in :
    y'=-(3 x^2 y^2 + y^3)/(2 x^3 y + 3 x y^2)
    Hi MaxJasper !
    Yes, the solution: (x^3)(y^2)+(y^3)x = C results in :
    y' = -(3 x^2 y^2 + y^3)/(2 x^3 y + 3 x y^2) = -(3x^2 * y + y^2) / (2x^3 + 3xy)
    Solving the modified version with (-) is rather easy. I let Shanter answer to your question since he found how to do it.
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  9. #9
    Senior Member MaxJasper's Avatar
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    Re: Nonlinear nonexact first order ODE

    Differentiate your solution and see that it is different from original diff eq.
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    Re: Nonlinear nonexact first order ODE

    Quote Originally Posted by MaxJasper View Post
    Differentiate your solution and see that it is different from original diff eq.
    I don't agree, the solution is allright.
    Publish on tbe forum what you did and we will locate your mistake.
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  11. #11
    Senior Member MaxJasper's Avatar
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    Re: Nonlinear nonexact first order ODE

    Your solution: (x^3)(y^2)+(y^3)x = C
    results in :

    y'=-(3 x^2 y^2 + y^3)/(2 x^3 y + 3 x y^2)

    that is different from original diff eq. Take a careful look.
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  12. #12
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    Re: Nonlinear nonexact first order ODE

    Why don't symplify ?
    (y) is a factor of the numerator and of the denominator.
    After simplification, compare to the diff. equation (the modified version with - , of course).
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    Re: Nonlinear nonexact first order ODE

    Hi MaxJasper !

    do yo agree now ?
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    Re: Nonlinear nonexact first order ODE

    Hey this might be a little late but if you write it out as
    Mdx + Ndy = 0
    To find the integrating factor I think I used
    (Nx - My)/M = A, int factor = e^integral A = y
    Note, A must be a function of y only.
    I don't have my work with me and I am typing this on my phone but I think this is all right. The rest is straight forward using everything you have.
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