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Math Help - second order non linear ODE

  1. #1
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    second order non linear ODE

    Does anyone can help me to solve this second order non linear ODE:

    y'' + (2/x)(y') - (1/2y)(y')(y') = K,

    y' = dy/dx

    y'' = dy'/dx

    y = y(x)

    I've already guess y=Ax^2 satisfy this equation, but I want to solve it analitically..

    I have transform the equation above with x=-(1/u) and f = ln H, that lead to a new equation:

    f'' + (1/2)(f')(f') = -(K/u^4)exp{-f}

    f = f(u)

    f' = df/du

    f'' = df'/du

    but still cannot find the solution analitically.

    Is y'' + (2/x)(y') - (1/2y)(y')(y') = K cannot be solved analitically?
    It seem strange to me that an an ODE with a simple subtitution solution y=Ax^2 don't have any analytical solution..

    Please help!
    Thanks before..
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  2. #2
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    Quote Originally Posted by ceramica View Post
    Does anyone can help me to solve this second order non linear ODE:

    y'' + (2/x)(y') - (1/2y)(y')(y') = K,

    y' = dy/dx

    y'' = dy'/dx

    y = y(x)

    I've already guess y=Ax^2 satisfy this equation, but I want to solve it analitically..

    I have transform the equation above with x=-(1/u) and f = ln H, that lead to a new equation:

    f'' + (1/2)(f')(f') = -(K/u^4)exp{-f}

    f = f(u)

    f' = df/du

    f'' = df'/du

    but still cannot find the solution analitically.

    Is y'' + (2/x)(y') - (1/2y)(y')(y') = K cannot be solved analitically?
    It seem strange to me that an an ODE with a simple subtitution solution y=Ax^2 don't have any analytical solution..

    Please help!
    Thanks before..
    The ODE

    y'' + (2/x)(y') - (1/2y)(y')(y') = K,

    is difficult to read - and what is the last term - \frac{1}{2y} y'^2 ?
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  3. #3
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    <br>
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  4. #4
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    Unhappy

    yes.. the ODE is:
    y''+ \frac{2}{x} y' - \frac{1}{2y} y'^2=K<br />
    just want to know whether this equation can be solved analitically or not.

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  5. #5
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    Quote Originally Posted by ceramica View Post
    yes.. the ODE is:
    y''+ \frac{2}{x} y' - \frac{1}{2y} y'^2=K
    just want to know whether this equation can be solved analitically or not.
    Are solutions with K = 0 of any interest?
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  6. #6
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    I've already solved it for K = 0<br />

    It is  y = (1 + \frac{a}{bx})^2<br />

    I need to know whether this ODE can (or cannot) be solved analitically, so I can do numerical solution as soon as possible.

    Thanks before.
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  7. #7
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    Forget to tell, K is a constant, in general, is not zero.
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