Question

求解幂级数的和函数

对和函数求导一次然后利用S（x）+S'（x）+1=e∧x然后解微分方程能够得出正确答案吗

Answer 1

求导后与原式相加即得 eˣ

解微分方程得 S(x)＝(eˣ－e⁻ˣ)/2

Answer 2

S(x) = ∑<n=0, ∞>x^(2n+1)/(2n+1)! = x + x^3/3! + x^5/5! + ......
S'(x) = ∑<n=0, ∞>x^(2n)/(2n)! = 1 + x^2/2! + x^3/4! + ......
S(x) + S'(x) = ∑<n=0, ∞>x^n/n! = e^x, S(0) = 0
通解 S(x) = e^(-∫dx)[∫e^xe^(∫dx)dx + C]
= e^(-x)[∫e^(2x)dx + C] = e^(-x)[(1/2)e^(2x) + C]
= (1/2)e^x + Ce^(-x),
S(0) = 0 代入， 得 C = -1/2，
S(x) = (1/2)e^x - (1/2)e^(-x) = sinhx