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)的解爲
y=C(x)e^(-∫P(x)dx) (C(x)是關於x的函數)
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).
