應用數學問題不知道如何解
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?
回答 (4)
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
設 y=Σ{n=0→∞} a_n (x-1)^(b+n).
what is the definition of a_n ?
收錄日期: 2021-05-04 01:43:04
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