一阶线性方程的公式

一阶线性微分方程dy/dx+P(x)y=Q(x)的通解公式应用“常数变易法”求解.

∵由齐次方程dy/dx+P(x)y=0

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

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

==>ln│y│=-∫P(x)dx+ln│C│ (C是积分常数)

==>y=Ce^(-∫P(x)dx)

∴此齐次方程的通解是y=Ce^(-∫P(x)dx)

于是，根据常数变易法，设一阶线性微分方程dy/dx+P(x)y=Q(x)的解为

y=C(x)e^(-∫P(x)dx) (C(x)是关于x的函数)

代入dy/dx+P(x)y=Q(x)，化简整理得

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是积分常数)

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

故一阶线性微分方程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是积分常数).