12 Ordinary Differential Equations IV

In this class,

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

1. Solve the system
   $$\begin{eqnarray*} x_1' & = & 2 x_1 + x_2 - 2 x_3 \\ x_2' & = & 3 x_1 -2 x_2 \\ x_3' & = & 3 x_1 + x_2 - 3 x_3 \end{eqnarray*}$$
   
   
   Find the general solution to this system in vector form and in terms of a fundamental matrix. Then find the vector of integration constants assuming that $\vec{x}(0)=(1,7,3)^{\rm T}$ and write $\vec{x}(t)$ for that case.

2. 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.

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 $\vec x$. No, you cannot apply initial conditions on the homogeneous solution.