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Math Help - [SOLVED] differential equation

  1. #1
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    [SOLVED] differential equation

    Determine the general solution of the differential equation

    y′ = (y/2x) − (xy)^3 .

    what method should i use for this
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by rajr View Post
    c)
    Determine the general solution of the differential equation

    y
    = (y/2x) (xy)^3 .

    what method should i use for this

    Rearranging, you get \frac{\,dy}{\,dx}-\frac{1}{2x}y=-x^3y^3

    This can be recognized as a Bernoulli Equation.

    Let z=y^{1-3}=y^{-2}\implies z^{-\frac{1}{2}}=y

    Thus, \frac{\,dy}{\,dx}=\frac{\,dy}{\,dz}\cdot\frac{\,dz  }{\,dx}=-\tfrac{1}{2}z^{-\frac{3}{2}}\frac{\,dz}{\,dx}

    Making the appropriate substitutions, we end up with:

    -\tfrac{1}{2}z^{-\frac{3}{2}}\frac{\,dz}{\,dx}-\frac{1}{2x}z^{-\frac{1}{2}}=-x^{3}z^{-\frac{3}{2}} which becomes the linear equation \frac{\,dz}{\,dx}+\frac{1}{x}z=2x^3

    Can you take it from here?
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  3. #3
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    thnk u

    is the integrating factor is lnx
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by rajr View Post
    thnk u

    is the integrating factor is lnx
    The integrating factor is {\color{red}e}^{\ln x}=x
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  5. #5
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    hi thx for ur reply

    i got z to be (2x^4/5) + (c/x)

    i got the final answer to be 1/[sqrt{(2x^4/5) + (c/x)}]

    is this right
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by rajr View Post
    hi thx for ur reply

    i got z to be (2x^4/5) + (c/x)

    i got the final answer to be 1/[sqrt{(2x^4/5) + (c/x)}]

    is this right
    You're correct!
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by rajr View Post
    hi thx for ur reply

    i got z to be (2x^4/5) + (c/x)

    i got the final answer to be 1/[sqrt{(2x^4/5) + (c/x)}]

    is this right
    yes, if by 2x^4/5 you mean \frac {2x^4}5
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  8. #8
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    Quote Originally Posted by Chris L T521 View Post
    You're correct!
    thank you for your help
    am done with my home work and the link which you gave for the DE was really helpful... it made me understand a lot better than other notes..

    tc
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  9. #9
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    Quote Originally Posted by Jhevon View Post
    yes, if by 2x^4/5 you mean \frac {2x^4}5
    yeah i got the same as what u mentioned thx
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