微分方程(x-2)y'=y+2(x-2)^3在初试条件 y|(x=1)=0 下的特解
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y' - y/(x - 2) = 2(x - 2)²
e^∫ -dx/(x - 2) = e^-ln|x - 2| = 1/(x - 2)，乘以方程两边
y'/(x - 2) - y/(x - 2)² = 2(x - 2)
[y/(x - 2)]' = 2(x - 2)
y/(x - 2) = ∫ (2x - 4) dx = x² - 4x + C
y = (x² - 4x + C)/(x - 2)
当x = 1，y = 0
0 = (1 - 4 + C)/(1 - 2) => C = 3
∴解是y = (x² - 4x + 3)/(x - 2) = x - 2 - 1/(x - 2)
e^∫ -dx/(x - 2) = e^-ln|x - 2| = 1/(x - 2)，乘以方程两边
y'/(x - 2) - y/(x - 2)² = 2(x - 2)
[y/(x - 2)]' = 2(x - 2)
y/(x - 2) = ∫ (2x - 4) dx = x² - 4x + C
y = (x² - 4x + C)/(x - 2)
当x = 1，y = 0
0 = (1 - 4 + C)/(1 - 2) => C = 3
∴解是y = (x² - 4x + 3)/(x - 2) = x - 2 - 1/(x - 2)
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