九、
\[
\frac{d\dot{x}}{dx} = \frac{-2\dot{x}^2 - \dot{x} - x + 2}{\dot{x}} = \frac{0}{0} \Rightarrow \begin{cases} x=2 \\ \dot{x}=0 \end{cases}
\]
显然只有一个奇点，不存在2个
判断奇点类型：
\[
\ddot{x} = -2\dot{x}^2 - \dot{x} - x + 2 = f
\]
\[
\Rightarrow \ddot{x} = \frac{\partial f}{\partial x}\bigg|_{xe} x + \frac{\partial f}{\partial \dot{x}}\bigg|_{xe} \dot{x} = -x - \dot{x}
\]
\[
\Rightarrow \lambda^2 + \lambda + 1 = 0 \Rightarrow \lambda_{1,2} = \frac{-1 \pm \sqrt{3}j}{2}
\]
为稳定焦点
十、
(1)
\[
G = \frac{k\left(\dfrac{s}{0.5}+1\right)}{s\left(\dfrac{s}{0.05}+1\right)\left(\dfrac{s}{10}+1\right)\left(\dfrac{s}{20}+1\right)}
\]
截止频率为5，由万能公式可得：
\[
20\lg\frac{k}{0.05} - 40\lg\frac{0.5}{0.05} - 20\lg\frac{5}{0.5} = 0 \Rightarrow k = 50
\]
\[
G_c = \frac{G}{G_0} = \frac{\dfrac{50\left(\dfrac{s}{0.5}+1\right)}{(0.05°+1)\left(\dfrac{s}{10}+1\right)\left(\dfrac{s}{20}+1\right)}}{\dfrac{50}{s\left(\dfrac{s}{2}+1\right)\left(\dfrac{s}{20}+1\right)}} = \frac{\left(\dfrac{s}{0.5}+1\right)\left(\dfrac{s}{2}+1\right)}{\left(\dfrac{s}{0.05}+1\right)\left(\dfrac{s}{10}+1\right)}
\]
(2)
\[
G_c = \frac{\left(\dfrac{s}{0.5}+1\right)\left(\dfrac{s}{2}+1\right)}{\left(\dfrac{s}{0.05}+1\right)\left(\dfrac{s}{10}+1\right)}
\]
91