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46. 特征方程 r^2+1 = 0, r = ±i,
对于 y''+y = x^2+3, 设特解为 y = ax^2+bx+c,
则 y' = 2ax+b, y'' = 2a, 2a + ax^2+bx+c = x^2+3,
则 a = 1, b = 0, c = 1, 特解为 y = x^2+1;
对于 y''+y = cosx, 设特解为 y = x(pcosx+qsinx),
则 y' = pcosx+qsinx + x(-psinx+qcosx) = (p+qx)cosx + (q-px)sinx,
y'' = qcosx - (p+qx)sinx - psinx + (q-px)cosx
= - (2p+qx)sinx + (2q-px)cosx
- (2p+qx)sinx + (2q-px)cosx + x(pcosx+qsinx) = cosx,
则 -2p = 0, 2q = 1, 则 q = 1/2, p = 0, 特解为 y = (1/2)xsinx
通解 y = C1cosx + C2sinx + x^2 + 1 + (1/2)xsinx
对于 y''+y = x^2+3, 设特解为 y = ax^2+bx+c,
则 y' = 2ax+b, y'' = 2a, 2a + ax^2+bx+c = x^2+3,
则 a = 1, b = 0, c = 1, 特解为 y = x^2+1;
对于 y''+y = cosx, 设特解为 y = x(pcosx+qsinx),
则 y' = pcosx+qsinx + x(-psinx+qcosx) = (p+qx)cosx + (q-px)sinx,
y'' = qcosx - (p+qx)sinx - psinx + (q-px)cosx
= - (2p+qx)sinx + (2q-px)cosx
- (2p+qx)sinx + (2q-px)cosx + x(pcosx+qsinx) = cosx,
则 -2p = 0, 2q = 1, 则 q = 1/2, p = 0, 特解为 y = (1/2)xsinx
通解 y = C1cosx + C2sinx + x^2 + 1 + (1/2)xsinx
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