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  • 1 # 小吶不帥但很實在

    一階線性微分方程dy/dx+P(x)y=Q(x)的通解公式應用“常數變易法”求解.

    ∵由齊次方程dy/dx+P(x)y=0

    ==>dy/dx=-P(x)y

    ==>dy/y=-P(x)dx

    ==>ln│y│=-∫P(x)dx+ln│C│ (C是積分常數)

    ==>y=Ce^(-∫P(x)dx)

    ∴此齊次方程的通解是y=Ce^(-∫P(x)dx)

    於是,根據常數變易法,設一階線性微分方程dy/dx+P(x)y=Q(x)的解為

    y=C(x)e^(-∫P(x)dx) (C(x)是關於x的函式)

    代入dy/dx+P(x)y=Q(x),化簡整理得

    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 (C是積分常數)

    ==>y=C(x)e^(-∫P(x)dx)=[∫Q(x)e^(∫P(x)dx)dx+C]e^(-∫P(x)dx)

    故一階線性微分方程dy/dx+P(x)y=Q(x)的通解公式是

    y=[∫Q(x)e^(∫P(x)dx)dx+C]e^(-∫P(x)dx) (C是積分常數).

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