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Homework Help: Help solving non-linear second order DE

  1. Feb 16, 2009 #1
    could anyone help me with solving this second order differential equation? im a noob on here so not sure how ya get the mathplayer stuff on... so the equation is of the form

    x'' +ax' + bx^n = 0

    (x^n means x to the power n, with a and b constants). i tried substituting x'=u to get a first order linear, but then got lost in the algebra of solving the linear DE with a integrating factor.. so is there a different method?

    thanks, Pete
    Last edited: Feb 16, 2009
  2. jcsd
  3. Feb 16, 2009 #2
    Don't you solve the associated quadratic?
  4. Feb 16, 2009 #3
    well substituing u=x' i get the equation

    u' + au = -bx^n

    which is of the form

    y' +ay = f(x)

    and so I multiply by the relevant integrating factor to solve, but the algebra gets very hard..

    i only know how to solve linear second order DE, and got the idea of substituting x'=u from other posts, and so dont know if it is the correct way to go about this problem...

    could you explain what the associated quadratic you are talking about is, as its probably something i havent yet come across...

    thanks, Pete
  5. Feb 16, 2009 #4
    Don't you just have

    x'' + ax' = -bx^n

    So first solve x'' + ax' = 0, do you know how to do that?

    Then assume a solution to solve the = -bx^n part?
  6. Feb 16, 2009 #5
    aah yes coz the sums of the solutions is a solution or summit like tht right??

  7. Feb 16, 2009 #6
    wait... i dont kno how to solve x'' + ax' = 0 haha
  8. Feb 16, 2009 #7
    Find the roots of q^2 + a*q = 0
  9. Feb 16, 2009 #8
    I'm also not sure why you think this is a nonlinear DE? I googled for "solving second order linear DEs", this is one of the links you might want to read: http://silmaril.math.sci.qut.edu.au/~gustafso/mab112/topic12/ [Broken]
    Last edited by a moderator: May 4, 2017
  10. Feb 16, 2009 #9
    wel the only second order DE i have experience of solving are linear constant coefficient ones, and this doesnt look like any ive come across before.. so i jst assumed this was non-linear... thanks for the link and help, i think i get it now..
  11. Feb 16, 2009 #10
    ive followed the link, and have come to a solution of

    y = A + B*exp(-ax) (1) for y'' + ay' = 0

    but then using the D and then Q operators i run into problems when trying to find the particalur integral of

    y'' + ay' = -by^n

    due to the A term in equation (1), which i cannot use the First Shift Theorem on...

    Any ideas? or have i gone wrong somewhere?
  12. Feb 17, 2009 #11


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    NoMoreExams, please, did you even look at the problem? x'' +ax' + bx^n = 0 is non-linear because of that 'x^n' term. None of your suggestions help here.
    Pete69, if you cannot solve a simple linear equation like x"+ ax= 0 you certainly cannot expect to solve a difficult non-linear problem like this! Looks like the blind leading the blind here. Pete69, where did you get this problem? Is it for a course?
  13. Feb 17, 2009 #12
    if you look at my other topic (a few below this one in this section, about free fall under gravity) you will see where it came from... iv only just come across linear constant coefficent second order DEs (homogenous and non-homogenous) so now know how to solve the x'' + ax = 0 part... but my course doesnt cover non-linear DEs, until next yr, or at all.. so have no clue how to solve them, but i was jst interested in furthering my knowledge..
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