Consider the linear system

$$\displaystyle \boldsymbol{ \vec{y}^{\,\prime} } = \left\lbrack \begin{array}{rr} -3 & -2 \\ 5 & 3 \end{array} \right\rbrack \boldsymbol{\vec{y}}.$$

1. Find the eigenvalues and eigenvectors for the coefficient matrix.

   $\lambda_1 =$

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

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

   , and $\lambda_2 =$

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

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

   
   
   
2. Find the real-valued solution to the initial value problem

   $$\left\lbrace \begin{array}{rcrcl} y_1^{\,\prime} & = & -3 y_1 - 2 y_2, & & y_1(0) = 1, \\ y_2^{\,\prime} & = & 5 y_1 + 3 y_2, & & y_2(0) = -5. \end{array} \right.$$
   Use $t$ as the independent variable in your answers.
   
   $y_1(t) =$
   $y_2(t) =$