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Non-homogeneous 2nd order diff eq involves power series

  1. May 9, 2006 #1
    I just need a hint or something to see where I start. I'm at a loss for a beginning.

    Consider the non-homogenous equation
    [tex]y'' + xy' + y = x^2 +2x +1[/tex]

    Find the power series solution about [tex]x=0[/tex] of the equation and express your answer in the form:

    [tex]y=a_0 y_1 + a_1 y_2 + y_p[/tex]

    where [tex]a_0[/tex] and [tex]a_1[/tex] are arbitrary constants. Give only the first three nonzero terms of each of the three series[tex]y_1[/tex],[tex]y_2[/tex], and [tex]y_p[/tex]

    Hint: Substitute [tex] y = \sum_{n=0}^{\infty}a_nx^{n}[/tex] and equate coefficients to find [tex]a_n[/tex], [tex]n = 2,3,4,5[/tex]
  2. jcsd
  3. May 9, 2006 #2


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    Homework Helper

    Do you know how to take the derivative of y in that form? If so, plug it in, and then try to rearrange the expression so that you have an infinite linear combination of powers of x that is equal to 0. Since the powers of x are linearly independent, all these coefficients must equal to zero, which will give you an expression for a_n in terms of a_n-1 and maybe a_n-2. This is called the Frobenius method, if you want to look online for a better explanation.
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