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Nonlinear Equation

  1. Jun 2, 2006 #1
    Hi,

    Can somebody guide me in getting solution for the following
    non-linear differential equation?

    u''=K/((a-u)^2) .....K is a constant

    Where u is a function of x and the boundary conditions are
    u(0)=0
    u(L)=0

    I need solution in form of 'u' as a function of 'x' with two constants, which are to be determined from the boundary conditions.

    Thanks
     
  2. jcsd
  3. Jun 2, 2006 #2

    arildno

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    Multiply with u', integrate, and see if you can do something more.
     
  4. Jun 3, 2006 #3

    arildno

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    To help you along a bit:
    [tex]u'u''=\frac{Ku'}{(a-u)^{2}}\to\frac{1}{2}(u')^{2}=C_{1}+\frac{K}{(a-u)}=\frac{C_{2}-C_{1}u}{a-u},C_{2}=C_{1}a+K[/tex]
    Or, we get:
    [tex]u'=\pm\sqrt{\frac{C_{3}-C_{4}u}{a-u}},C_{3}=2C_{2},C_{4}=2C_{1}[/tex]

    Rewriting, and integrating, we have:
    [tex]\int\sqrt{\frac{a-u}{C_{3}-C_{4}u}}du=x+C_{5}[/tex]
    You may crack the integral at your left-hand side using the substitution:
    [tex]v=\sqrt{\frac{a-u}{C_{3}-C_{4}u}}[/tex]
    since we have:
    [tex]u=\frac{a-C_{3}v^{2}}{1-C_{4}v^{2}}[/tex]
    Thus, the problem is resolved to partial fractions decomposition, and an implicit formula of the solution function.
    You might well encounter some non-trivial conditions placed on K and a when imposing the boundary conditions.

    Good luck! :smile:
     
    Last edited: Jun 3, 2006
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