欧拉方程。设 x = e^t, 则
dy/dx = (dy/dt)/(dx/dt) = e^(-t)dy/dt = (1/x)dy/dt
d^2y/dx^2 = d[(1/x)dy/dt]/dx = -(1/x^2)dy/dt + (1/x)(d^2y/dt^2)/(dx/dt)
= -(1/x^2)dy/dt + (1/x^2)(d^2y/dt^2)
微分方程化为
d^2y/dt^2 - 2dy/dt - 8y = e^(2t)
特征方程 r^2 - 2r - 8 = 0, r = 4, -2
设特解 y = ae^(2t), 代入解得 a = -1/8, 特解 y = -(1/8)e^(2t)
通解 y = C1e^(4t) + C2e^(-2t) - (1/8)e^(2t)
即 y = C1x^4 + C2/x^2 - (1/8)x^2
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