工程數學之求收斂區間及解題

Q1:
1. xy"+y'=0 => x^2y"+xy'=0 is an Euler's Eq.
aux. eq. m(m-1)+m=0, m=0,0
=> y= a x^0 + bx^0 lnx = a+b lnx
2. Power series mathod
xy"+y'=0, x=0 is an irregular singular point,
Let y=Σ[0~∞] a(n) x^(n+r) be a solution of xy"+y'=0
thus, Σ[0~∞] [(n+r)(n+r-1)a(n)+(n+r)a(n)]x^(n+r-1) =0 ---(A)
when n=0 (radical eq.) r(r-1)+r=0, r=0, 0( multiple root)
By Frobenius' theorem:
case 1: r=0
From (A): n(n-1)a(n)+na(n)=0, n=0,1,2,...
=> a(0) is arbitrary. a(1)=a(2)=...=0
so, y=a(0) ( a constant fn.)
Set y1= 1
case 2:
y2= y1 lnx+Σ[0~∞] a(n)x^(n+0) = lnx +Σ[0~∞] a(n)x^n
Putting into xy"+xy'=0, then
Σ[0~∞] [n(n-1)+n]a(n)x^(n-1) =0
=>a(0) is arbitrary, a(1)=a(2)=... =0 => y2= lnx +a(0)
so, y=a+b lnx
Q2:
By the ratio test:
lim(n->∞) | a(n+1)/a(n) | = | (x-1)^3 / 5 |
case 1: If | (x-1)^3 | < 5, then the infinite series is conv.
=> | x-1 | < 5^(1/3)
case 2: If | (x-1)^3 | > 5, then the series is div.
case 3: If (x-1)^3 = +/- 5, then the series equals
Σ[0~∞] 1 or Σ[0~∞] (-1)^m both are div.
=> the interval of conv. is | x-1 | < 5^(1/3) or ( 1-5^(1/3), 1+5^(1/3) )