设f(x)为可导函数,且满足f(x)=∫（上限X下线1）f(t)/tdt+(x-1)e^x求f(x)

来源：学生作业帮编辑：搜搜考试网作业帮分类：数学作业时间：2024/08/16 18:58:45

f(x) = ∫(1,x) f(t)/t dt + (x - 1)e^x
f'(x) = f(x)/x * dx/dx + e^x * (1 - 0) + (x - 1) * e^x
f'(x) = f(x)/x + xe^x
==> y' = y/x + xe^x
==> y' - y/x = xe^x、e^(∫ - 1/x dx) = e^(- lnx) = 1/x、将1/x乘以方程的两边.
==> y'/x - y/x² = e^x
==> (y/x)' = e^x
==> y/x = C + e^x
==> y = Cx + xe^x
f(x) = Cx + xe^x
令x = 1、f(1) = 0 ==> 0 = C + e ==> C = - e
所以f(x) = xe^x - e * x

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