设f(x)为连续函数且满足∫f(x-t)etdt=sinx,求f（x）

Question

设f(x)为连续函数且满足∫f(x-t)etdt=sinx,求f（x）

Answer 1

f(x) = sinx - ∫(0~x) (x - t) f(t) dt
= sinx - x∫(0~x) f(t) dt + ∫(0~x) tf(t) dt,之后两边对x求导
f'(x) = cosx - [x' · ∫(0~x) f(t) dt + x · f(x)] + xf(x)
f'(x) = cosx - ∫(0~x) f(t) dt,两边再对x求导
f''(x) = - sinx - f(x)
==> y'' + y = - sinx,解微分方程
特征方程：r² + 1 = 0 => r = ±i
y = Acosx + Bsinx
令特p = x · (Acosx + Bsinx) = Axcosx + Bxsinx
p'' = - Axcosx - 2Asinx + 2Bcosx - Bxsinx,代入微分方程中
p'' + p = - sinx
(- Axcosx - 2Asinx + 2Bcosx - Bxsinx) + (Axcosx + Bxsinx) = - sinx
- 2Asinx + 2Bcosx = - sinx
解得A = 1/2,B = 0
p = (1/2)xcosx
通解为y = (1/2)xcosx + Acosx + Bsinx
所以f(x) = (1/2)xcosx + Acosx + Bsinx,其中A和B都是任意常数