應用數學問題不知道如何解

Question

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?

Answer 1

最佳解答

(2)

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：

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpa...

參考資料：

formula from http://en.wikipedia.org/wiki/Taylor_series#Example... + my wisdom of maths

Answer 2

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

Answer 3

設 y=Σ{n=0→∞} a_n (x-1)^(b+n).

Answer 4

what is the definition of a_n ?