The problem involves solving a system of linear differential equations with initial conditions:

$\frac{dx}{dt} = 3x + 8y \quad \text{and} \quad \frac{dy}{dt} = -x - 3y$

with the initial conditions $x(0) = 6$ and $y(0) = -2$.

Solution Outline

This is a system of first-order linear differential equations, which can be solved using matrix methods or decoupling through eigenvalues and eigenvectors.

1. Write the system in matrix form: $\frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 & 8 \\ -1 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$

2. Find the eigenvalues and eigenvectors of the coefficient matrix: $A = \begin{pmatrix} 3 & 8 \\ -1 & -3 \end{pmatrix}$

3. Solve the characteristic equation: $\text{det}(A - \lambda I) = 0$

4. Use the eigenvalues and eigenvectors to express the general solution.

5. Apply initial conditions $x(0) = 6$ and $y(0) = -2$ to find the specific solution.

Let me compute this step by step.The eigenvalues of the matrix $A$ are ( \lambda_