Answer

问题及解答

求解微分方程 $x''(t)-1000(1-x^2)x'(t)+x=0$.

Posted by haifeng on 2018-05-04 08:29:06 last update 2018-05-07 10:25:31 | Edit | Answers (1)

求解微分方程

\[
\begin{cases}
\dfrac{d^2 x}{dt^2}-1000(1-x^2)\dfrac{dx}{dt}+x(t)=0,\\
x(0)=0,\quad x'(0)=1.
\end{cases}
\]
 

 


[Hint]

将高阶微分方程转换为与之等价的一阶微分方程组.

 

 

References:

《数学建模与数学实验》(第四版)P.131

 

hello

1

Posted by haifeng on 2018-05-04 08:36:16

令 $u=x(t)$, $v=u'(t)=x'(t)$. 则 $v'(t)=x''(t)$. 于是原方程变为

\[
\begin{cases}
v'-1000(1-u^2)v+u=0,\\
v=u',\\
u(0)=0,\quad v(0)=1.
\end{cases}
\]