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Initial value problem Euler equation

  1. Nov 1, 2011 #1

    Find y as a function of x:

    x^2 y'' + 8 x y' - 18 y = x^8

    y(1)=3, y'(1)=2

    Attempted solution:

    I found the general equation to be Ax^(-9)+Bx^2+Cx^8.
    However when I try to solve the initial value problem for this equation I have 3 unknowns.
  2. jcsd
  3. Nov 1, 2011 #2

    I like Serena

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    Homework Helper

    Welcome to PF, lisa92! :smile:

    Did you substitute your solution in the DE?
    If you do, you'll find you have a 3rd equation.
  4. Nov 1, 2011 #3


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    (x2D2+8 x D-18)=(xD+9)(xD-2)
    or change variables u=x2 y

    your general solution is extraneous check it in the equation to eliminate
  5. Nov 1, 2011 #4


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    Re: Help for this equation?

    That's an "Euler type" equationl. The change of variable x= ln(t) changes it to a differential equation with constant coefficients which may be simpler.

    If you do not wish to do it that way, what did you get as a solution to the associated homogeneous equation?
  6. Nov 1, 2011 #5


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    Re: Help for this equation?

    Check out Euler-Cauchy equations at the bottom of this link:


    This will help you obtain the solution to the homogeneous equation.

    The particular solution will probably be y = k * x^8 + y (homogeneous)
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