(Solved): Advanced Ordinary Differential Equations Question:Thorough Detail Please! Describe the behavior of t ...

Describe the behavior of the following system Sor all possible values of \( m, k \), and \( b \) (where \( m>0, k \geq 0 \), and \( b \geq 0 \), and all possible initial conditions (i.e. \( x(0) \) and \( \dot{x}(0) \) ). \[ \begin{array}{ll} & m \ddot{x}+b \dot{x}+k x=0 \quad \text { (DampeciOscilletor) } \\ x_{1}=x & \dot{x}_{1}=x_{2} \\ x_{2}=\dot{x} & \dot{x}_{2}=-\frac{k}{m} x_{1}-\frac{b}{m} x_{2}=\left[\begin{array}{cc} 0 & 1 \\ -\frac{b}{m} & -\frac{b}{m} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \\ \vec{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \quad \vec{x}=A \vec{x} \text { for } A=\left[\begin{array}{cc} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{array}\right] \end{array} \]