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f(x) = sinx - x∫<0, x>f(t)dt + ∫<0, x>tf(t)dt, f(0) = 0;
对 x 求导,f'(x) = cosx - ∫<0, x>f(t)dt - xf(x)+ xf(x)
f'(x) = cosx - ∫<0, x>f(t)dt, f'(0) = 1.
对 x 再求导,得 f''(x) = -sinx - f(x), f'(0) = 1.
即得 微分方程 y''+ y = -sinx , y(0) = 0, y'(0) = 1
特解 y = (1/2)xcosx,
通解 y = Acosx + Bsinx + (1/2)xcosx
y(0) = 0, A = 0
y'(0) = 1, 得 B = 1/2
则 y = f(x) = (1/2)(sinx + xcosx)
对 x 求导,f'(x) = cosx - ∫<0, x>f(t)dt - xf(x)+ xf(x)
f'(x) = cosx - ∫<0, x>f(t)dt, f'(0) = 1.
对 x 再求导,得 f''(x) = -sinx - f(x), f'(0) = 1.
即得 微分方程 y''+ y = -sinx , y(0) = 0, y'(0) = 1
特解 y = (1/2)xcosx,
通解 y = Acosx + Bsinx + (1/2)xcosx
y(0) = 0, A = 0
y'(0) = 1, 得 B = 1/2
则 y = f(x) = (1/2)(sinx + xcosx)
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