To solve this problem, we need to determine the eigenvalues and eigenvectors of the matrix $A = \begin{bmatrix} 1 & -3 \\ 2 & -4 \end{bmatrix}.$

Step 1: Finding the Eigenvalues ($d_1$ and $d_2$)

The eigenvalues are found by solving the characteristic equation of the matrix $A$: $\det(A - \lambda I) = 0.$ This gives: $\det \begin{bmatrix} 1 - \lambda & -3 \\ 2 & -4 - \lambda \end{bmatrix} = 0.$ Expanding the determinant, we get: $(1 - \lambda)(-4 - \lambda) - (-3)(2) = 0.$ Simplifying, $\lambda^2 + 3\lambda - 2 = 0.$ Solving this quadratic equation yields the eigenvalues $d_1$ and $d_2$.

Step 2: Finding the Eigenvectors ($\mathbf{v}_1$ and $\mathbf{v}_2$)

For each eigenvalue $d_i$, we substitute it back into $(A - d_i I)\mathbf{v} = 0$ to find the corresponding eigenvector $\mathbf{v}_i$.

Step 3: Verifying Initial Condition

After finding the eigenvalues and eigenvectors, we construct the solution: $\mathbf{x}(t) = e^{d_1 t} \mathbf{v}_1 + e^{d_2 t} \mathbf{v}_2.$ We then apply the initial condition $\mathbf{x}(0) = \begin{bmatrix} -3 \\ -3 \end{bmatrix}$ to determine the specific constants needed.

I'll perform these calculations to find $\begin{bmatrix} d_1 \\ d_2 \end{bmatrix}$, $\mathbf{v}_1$, and $\mathbf{v}_2$.The calculations yield the following results:

1. Eigenvalues ($d_1$ and $d_2$): $\begin{bmatrix} d_1 \\ d_2 \end{bmatrix} = \begin{bmatrix} -2 \\ -1 \end{bmatrix}$ where $d_1 = -2$ and $d_2 = -1$ (with $d_1 \le d_2$ as required).

2. Eigenvector $\mathbf{v}_1$ corresponding to $d_1 = -2$: $\mathbf{v}_1 = \begin{bmatrix} 0.7071 \\ 0.7071 \end{bmatrix}$

3. Eigenvector $\mathbf{v}_2$ corresponding to $d_2 = -1$: $\mathbf{v}_2 = \begin{bmatrix} 0.8321 \\ 0.5547 \end{bmatrix}$

These values satisfy the given conditions for the differential equation system.

Would you like further details on any part of this solution or have any questions?

Here are five related questions for further exploration:

1. How do we determine the general solution to a system of differential equations?
2. Why are eigenvalues important in solving linear differential equations?