【神秘】一阶线性常微分方程解法

### 0. 转化为标准形式 $y'(x)+p(x)y(x)=q(x)$ ### 1. 两边同乘神秘函数$u(x)=e^{\int p(x)dx}$ 此处$p(x)u(x)=u'(x)^*$ $y'(x)u(x)+p(x)u(x)y(x)=q(x)u(x)$ $y'(x)u(x)+u'(x)y(x)=q(x)u(x)$ $(y(x)u(x))'=q(x)u(x)$ ### 2. 两边积分 $y(x)u(x)=\int q(x)u(x)dx$ 整理得$y(x)=\frac{\int q(x)e^{\int p(x)dx}}{e^{\int p(x)dx}}$ #### *. 如何构造出满足$p(x)u(x)=u'(x)$的$u(x)$ $p(x)u(x)=\frac{du(x)}{dx}$ $p(x)dx=\frac{du(x)}{u(x)}$ 两边积分得$\int p(x)dx=lnu(x)$ 即 $u(x)=e^{\int p(x)dx}$ #### **记得+C