12 Ordinary Differential Equations IV

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Given the system
   
   $$\dot {\vec x} = A \vec x \qquad A = \left(\begin{array}{cc} 0 & 5 -1 & -2 \end{array} \right)$$
   
   
   Find the general solution to this system in vector form and in terms of a fundamental matrix. Complex solutions not allowed.

2. Solve the system and initial condition
   
   $$\dot {\vec x} = A \vec x \quad \vec x(0) = \vec x_0 \qquad ... ...x_0 = \left( \begin{array}{c} 9 1 1 \end{array} \right)$$
   
   
   Give a fundamental matrix. Clean up the final $\vec x$.

3. Solve the inhomogeneous system and initial condition
   
   $$\dot {\vec x} = A \vec x + \vec g \quad \vec x(0) = \vec x... ...0 = \left( \begin{array}{c} 5 11 -2 \end{array} \right)$$
   
   
   Use variation of parameters. Clean up the final $\vec x$.