Consider the initial value problem

$$\displaystyle \begin{array}{rcll} y_1^{\,\prime} & = & 3 y_2, & \ y_1(0) = 1, \\ y_2^{\,\prime} & = & -3 y_1 + 6 y_2, & \ y_2(0) = 2. \end{array}$$

1. Find the eigenvalue $\lambda$, an eigenvector $\boldsymbol{\vec{v}_1}$, and a generalized eigenvector $\boldsymbol{\vec{v}_2}$ for the coefficient matrix of this linear system.

   $\lambda =$

   , $\boldsymbol{\vec{v}_1} =$

   $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

   , $\boldsymbol{\vec{v}_2} =$

   $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

   
   
   
2. Find the most general real-valued solution to the linear system of differential equations. Use $c_1$ and $c_2$ to denote arbitrary constants, and enter them as c1 and c2.
   $y_1(t) =$
   $y_2(t) =$
   
   
3. Solve the original initial value problem.
   $y_1(t) =$
   $y_2(t) =$