【神秘】一阶线性常微分方程解法

· · 个人记录

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