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Find the eigenvalues and eigenvectors of the matrix $\displaystyle \left\lbrack \begin{array}{rr} -6 & -7 \\ 14 & 15 \end{array} \right\rbrack.$

$\lambda_1 =$

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

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

, and $\lambda_2 =$

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

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

Solve the system of differential equations $\displaystyle \boldsymbol{\vec{x}^{\, \prime}} = \left\lbrack \begin{array}{rr} -6 & -7 \\ 14 & 15 \end{array} \right\rbrack \boldsymbol{\vec{x}}$ satisfying the initial conditions $\displaystyle \left\lbrack \begin{array}{r} x_1(0) \\ x_2(0) \end{array} \right\rbrack = \left\lbrack \begin{array}{r} -4 \\ 6 \end{array} \right\rbrack.$

$x_1(t) =$
$x_2(t) =$

You can earn partial credit on this problem.