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).