Differential Equations

$求微分方程：\quad {\frac{dy}{dx}} + P(x)y = Q(x)\quad 的通解$

 

$\quad {\frac{dy}{dx}} + P(x)y = Q(x)$

$\quad \Rightarrow dy + yP(x)dx = Q(x)dx \quad \quad \quad (1)$

 

$\quad \forall P(x)，\exists R(x)，使得：$

$\quad R(x)dy + yP(x)R(x)dx = d(R(x)y) = R(x)dy + yR'(x)dx$

$\quad \Rightarrow P(x)R(x) = R'(x)$

$\quad \Rightarrow P(x) = ln'R(x)$

$\quad \Rightarrow R(x) = e^{\int P(x)dx}$

 

$\quad 对(1)式两边同乘R(x)，则有：$

$\quad R(x)dy + yP(x)R(x)dx = d(R(x)y) = Q(x)R(x)dx$

$\quad 两边求积：R(x)y = \int Q(x)R(x)dx + C$

$\quad \therefore y = (\int Q(x)R(x)dx + C) / R(x) = e^{-\int P(x)dx}(\int Q(x)e^{-\int P(x)dx}dx + C)$