Question

高数问题，一阶线性微分方程中提到的常数变易法，它的定义是什么，它是在什么问题中应用的

Answer 1

自然是一阶线性方程之中用到的
对于y' + P(x)y = Q(x)
先找出齐次方程的解
y' + P(x)y = 0
解为y = Ce^[- ∫ P(x) dx]
令C = C(x)
可再设y = C(x)e^[- ∫ P(x) dx]，这是常数变易法。
y' = C'(x)e^[- ∫ P(x) dx] - C(x)e^[- ∫ P(x) dx] * P(x)
代入非齐次方程中
C'(x)e^[- ∫ P(x) dx] - P(x)C(x)e^[- ∫ P(x) dx] + P(x)C(x)e^[- ∫ P(x) dx] = Q(x)
C'(x)e^[- ∫ P(x) dx] = Q(x)
C'(x) = Q(x)e^[∫ P(x) dx]
C(x) = ∫ {Q(x)e^[∫ P(x) dx]} dx + C
y = e^[- ∫ P(x) dx] * {∫ {Q(x)e^[∫ P(x) dx]} dx + C}，此乃非齐次方程的通解

例如解y' + xy = x
y' + xy = 0的解为y = Ce^(- ∫ x dx) = Ce^(- x²/2)
令y' + xy = x²的解为y = C(x)e^(- x²/2)
y' = C'(x)e^(- x²/2) - xC(x)e^(- x²/2)
代入y' + xy = x
C'(x)e^(- x²/2) - xC(x)e^(- x²/2) + xC(x)e^(- x²/2) = x
C'(x)e^(- x²/2) = x
C'(x) = xe^(x²/2)
C(x) = (e^(x²/2) + C
代入y = C(x)e^(- x²/2)得
y = [e^(x²/2) + C]e^(- x²/2) = 1 + Ce^(- x²/2)

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