问答题(2021年经济数学Ⅲ

设n为正整数,y=yn (x)是微分方程xy' - (n+1)y=0满足条件yn(1)=1/n(n+1)的解.

(1) 求yn (x);

(2) 求级数yn(x)的收敛域及和函数.

答案解析

(1) xy' - (n+1)y = 0 化简得 = (n+1),两边积分得ln⁡|y| = (n+1) ln⁡|x| + C0,化简得y = Cxn+1.∵ yn(1) = ,∴C = ,∴ yn (x) = xn+1.(2) 令<1,得-1<x<1.x=1时,收敛;x=-1...

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