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Second order nonlinear differential equation

  1. May 13, 2012 #1
    Is it possible to solve the following differential equation analytically?

    y''(x) = A - B [exp(y(x)/C) - 1]

    where A, B and C are constants.

    Thank you.........
  2. jcsd
  3. May 15, 2012 #2


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    Science Advisor

    that can be simplified to
    [tex]y'= A- Be^{y/C}- B= (A- B)- Be^{y/C}= A'- Be^{y/C}[/tex]
    with A- B= A'
    and if you let z= y/C, y= Cz, so y''= Cz'' and the equation becomes
    Cz''= A'- Be^z or z''= A"- B'e^z
    with A"= A'/C and B'= B/C.

    Since the independent variable, x, does not appear in that, let u= z' so that z''= u'= (du/dz)(dz/dx)= u(du/dz) and the equation becomes
    u(du/dz)= A"- B'e^z so udu= (A"- B'e^z)dz and, integrating,
    (1/2)u^2= A"z- B'e^z+ C

    So u= dz/dt= sqrt(2(A"z- B'e^z+ C))

    dt= dz/sqrt(2(A"z- B'e^z+ C))

    That looks like it cannot be integrated analytically but at least it is set up for a direct numerical integration.
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