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捲毛 發問時間: 科學數學 · 1 0 年前


Consider following diffrential equation:

(lnx) d^2y/dx^2 +1/2 (dy/dx) +y=0

(1)Determine the first four nonzero terms in the sereis Σ(n=0-->∞)an (x-1)^(b+n)?

(2)Assuming x>1 what would you expect the radius of convergence of the series solutions you get in (1) to be?

4 個解答

  • 1 0 年前


    As same as the radius of convergence of the taylor series of ln x(我估計)

    2011-01-12 22:51:55 補充:



    formula from + my wisdom of maths

  • 1 0 年前

    The differential equation (lnx) d^2y/dx^2 +1/2 (dy/dx) +y=0 (*)

    is recognized as a 2nd order linear equation for y=y(x); the point x=1 is a regular singular point of it [leave it to you to check].

    2011-01-06 04:19:59 補充:

    Hence in some neighborhood of x=1, there exists a series solution of the form y(x)= Σ(n=0-->∞)an (x-1)^(b+n), for 1<1+R, where b is a real number satisfying the so-called indicial equation of (*), and R is the radius of convergence of this series------The Frobenius Theorem.

    2011-01-06 04:20:44 補充:

    Therefore, in order to answer

    (1) : approximate lnx by its Taylor expansion [for 5 or 6 terms] about x=1 and sub y(x)= Σ(n=0-->∞)an (x-1)^(b+n) into (*) to finish the routine calculation. [ you are supposed to find b and the recurrence relation for a_n's]

    2011-01-06 04:21:12 補充:

    (2) R=1, which is the distance from x=1[the center of the series] to the point x=0[the only other singular point of (*) ---- directly from The Frobenius Theorem.

    2011-01-06 04:23:57 補充:

    更正 y(x)= Σ(n=0-->∞)an (x-1)^(b+n), for 1<1+R -->

    y(x)= Σ(n=0-->∞)an (x-1)^(b+n), for 1<1+R

  • 1 0 年前

    設 y=Σ{n=0→∞} a_n (x-1)^(b+n).

  • 1 0 年前

    what is the definition of a_n ?