The system of differential equations given is:

$\frac{dx}{dt} = 2x - 8y$ $\frac{dy}{dt} = x + 6y$ with initial conditions $x(0) = 4$ and $y(0) = 1$.

This system can be solved using methods for solving coupled first-order linear differential equations, such as:

1. Matrix method (diagonalization of matrices).
2. Substitution and solving each equation independently.
3. Eigenvalue and eigenvector approach for linear systems.

Here, I will use the matrix method. We can represent the system in matrix form as:

$\frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & -8 \\ 1 & 6 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$

The solution involves finding the eigenvalues and eigenvectors of the coefficient matrix:

$A = \begin{pmatrix} 2 & -8 \\ 1 & 6 \end{pmatrix}$

Let me first calculate the eigenvalues and eigenvectors of this matrix and then proceed with the general solution.The eigenvalues of the matrix $A$ are:

$\lambda_1 = 4 + 2i, \quad \lambda_2 = 4 - 2i$

The corresponding eigenvectors are:

$v_1 = \begin{pmatrix} 0.9428 \\ -0.2357 - 0.2357i \end{pmatrix}, \quad v_2 = \begin{pmatrix} 0.9428 \\ -0.2357 + 0.2357i \end{pmatrix}$

With these eigenvalues and eigenvectors, the general solution for the system will be of the form:

$\begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = c_1 e^{(4 + 2i)t} \begin{pmatrix} 0.9428 \\ -0.2357 - 0.2357i \end{pmatrix} + c_2 e^{(4 - 2i)t} \begin{pmatrix} 0.9428 \\ -0.2357 + 0.2357i \end{pmatrix}$

Next, we can simplify this by converting the complex exponentials into real functions using Euler's formula. Let me compute the full real solution and apply the initial conditions $x(0) = 4$ and $y(0) = 1$.To continue solving the system and apply the initial conditions, the complex solution can be written in real terms by using Euler's formula $e^{i \theta} = \cos(\theta) + i \sin(\theta)$. This converts the complex exponentials into sine and cosine functions.

The general real solution will take the form:

$x(t) = e^{4t} \left[ c_1 \cos(2t) + c_2 \sin(2t) \right]$ $y(t) = e^{4t} \left[ c_3 \cos(2t) + c_4 \sin(2t) \right]$

Now, applying the initial conditions $x(0) = 4$ and $y(0) = 1$, we solve for the constants $c_1, c_2, c_3, c_4$.

Let me compute these constants.The constants obtained from the initial conditions are:

$c_1 = 4, \quad c_3 = 1$

Thus, the real form of the solution becomes:

$x(t) = e^{4t} \left[ 4 \cos(2t) + c_2 \sin(2t) \right]$ $y(t) = e^{4t} \left[ 1 \cos(2t) + c_4 \sin(2t) \right]$

To fully determine the constants $c_2$ and $c_4$, we need the derivative conditions for $x'(0)$ and $y'(0)$. Would you like to proceed with this approach or have additional conditions?

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Would you like more details on the steps so far, or do you have any questions?

Follow-up questions:

1. How are eigenvalues used in solving differential systems?
2. Can you explain how Euler’s formula helps transform complex solutions into real solutions?
3. What would the solution look like without eigenvalue analysis?
4. How can we solve such systems graphically?
5. What happens if initial conditions are complex?

Tip:

For coupled systems, always check for symmetry in the coefficient matrix to explore easier solution methods.