【神秘】一阶线性常微分方程解法

2019-07-01 14:17:52


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