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Homework Help: Linear 2nd order ODE?

  1. Dec 28, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex]y''x^{2}[/tex] + 4xy' -y = ln(x)

    3. The attempt at a solution
    -I considered the quadratic characteristic equation, but it wont work because of the x^2
    -I also tried variation of parameters.
    so i have v = y' and v'=y'' but i have no idea what i would sub when I get to y.

    any ideas?
  2. jcsd
  3. Dec 28, 2009 #2


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    Hi kyin01! :smile:

    Try changing x

    look for a u(x) such that the equation becomes something simple in dy/du and d2y/du2. :wink:
  4. Dec 28, 2009 #3
    I think the standard way is to first find the solutions of the assoc. homogeneous DE [tex]y''x^{2}[/tex] + 4xy' -y =0 using Cauchy Euler DE by letting [tex]y=x^{m}[/tex].

    Then use variation of parameter etc. to find the particular solution.
  5. Dec 29, 2009 #4


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    Yes, although that can make the calculations pretty complicated. As tiny-time suggested, the substitution u= ln(x) will change any "Euler-type" or "Equipotential" equation to a very simple equation with constant coefficients.

    I rather suspect that if kyin01 were to check his recent notes or textbook where this problem is given, he would find a discussion of "Euler-type" or "Equipotential" equations. (The name varies from text book to text book.)
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