Consider the system of differential equations

$$\boldsymbol{\vec{y}^{\,\prime}} = \left\lbrack \begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array} \right\rbrack \boldsymbol{\vec{y}}, \ \ \ \boldsymbol{\vec{y}}(0) = \left\lbrack \begin{array}{r} -11 \\ 5 \end{array} \right\rbrack.$$
Verify that

$$\boldsymbol{\vec{y}}(t) = c_1 e^{4 t} \left\lbrack \begin{array}{r} 1 \\ 1 \end{array} \right\rbrack + c_2 e^{-2 t} \left\lbrack \begin{array}{r} 1 \\ -1 \end{array} \right\rbrack.$$
is a solution to the system of differential equations for any choice of the constants $c_1$ and $c_2$. Find values of $c_1$ and $c_2$ that solve the given initial value problem. (According to the uniqueness theorem, you have found the unique solution of $\boldsymbol{\vec{y}^{\,\prime}} = P \boldsymbol{\vec{y}}$,   $\boldsymbol{\vec{y}}(0) = \boldsymbol{\vec{y}_0}$).