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Math Help - homogenous solution of diff. ekv

  1. #1
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    homogenous solution of diff. ekv

    Hello

    I have a linear differential ekvation
    d2u/dr2 + 1/r*du/dr -u/r^2=-(1-v)*pw
    I want to have the homogenous solution to this,
    I know I should put the right side equal to zero.
    But after that my mind is blank...
    I think that u(0)=0
    Please help

    Thanks.
    Last edited by danielel; February 20th 2009 at 02:24 AM.
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  2. #2
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    Quote Originally Posted by danielel View Post
    Hello

    I have a linear differential ekvation
    d2u/dr2 + 1/r*du/dr -u/r^2=-(1-v)*pw
    I want to have the homogenous solution to this,
    I know I should put the right side equal to zero.
    But after that my mind is blank...
    I think that u(0)=0
    Please help

    Thanks.
    For the homogeneous problem

    \frac{d^2u}{dr^2} + \frac{1}{r} \frac{du}{dr} - \frac{u}{r^2} = 0

    or

    r^2 \frac{d^2u}{dr^2} + r \frac{du}{dr} - u = 0

    if you seek solutions of the form u = r^m then we find that m satisfies m(m-1) + m - 1 = 0 (typically called the characteristic equation) and this gives  m = -1, 1 and the solution

    u =c_1 r + \frac{c_2}{r}

    although I think you have a problem with your IC.
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  3. #3
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    How did you go from with m=1.1 to the solution
    ?
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  4. #4
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    Quote Originally Posted by danielel View Post
    How did you go from with m=1.1 to the solution
    ?
    Since m = -1,\;\; m = 1 then two solutions are u = r^1,\;\;\text{and}\;\; u = r^{-1} and since the ODE is linear then any linear combination is also a solution giving u = c_1 r + \frac{c_2}{r}
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