To solve a system of differential equations using the matrix method, follow these steps:

Example System of Differential Equations:

Suppose you are given the system: $\frac{dx}{dt} = ax + by$ $\frac{dy}{dt} = cx + dy$

This can be written in matrix form as: $\frac{d\mathbf{X}}{dt} = A\mathbf{X},$ where: $\mathbf{X} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}.$

Step-by-Step Solution:

1. Write the system in matrix form: Express the system as $\frac{d\mathbf{X}}{dt} = A\mathbf{X}$, where $A$ is the coefficient matrix.

2. Find the eigenvalues of $A$: Solve the characteristic equation $\det(A - \lambda I) = 0$ to find the eigenvalues $\lambda_1$ and $\lambda_2$.

3. Find the eigenvectors of $A$: For each eigenvalue $\lambda_i$, solve $(A - \lambda_i I)\mathbf{v} = 0$ to find the eigenvectors $\mathbf{v}_i$.

4. Construct the general solution: The solution to the system can be written as: $\mathbf{X}(t) = C_1 e^{\lambda_1 t} \mathbf{v}_1 + C_2 e^{\lambda_2 t} \mathbf{v}_2,$ where $C_1$ and $C_2$ are constants determined by initial conditions, and $\mathbf{v}_1, \mathbf{v}_2$ are the eigenvectors.

5. Apply initial conditions: Substitute the initial conditions (e.g., $\mathbf{X}(0) = \begin{bmatrix} x_0 \\ y_0 \end{bmatrix}$) to solve for $C_1$ and $C_2$.

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Example:

Suppose $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.

1. Find eigenvalues: $\det(A - \lambda I) = \det\begin{bmatrix} 2-\lambda & 1 \\ 1 & 2-\lambda \end{bmatrix} = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = 0.$ Solving, $\lambda = 1, 3$.

2. Find eigenvectors: For $\lambda = 1$: $(A - I)\mathbf{v} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = 0 \implies v_1 = -v_2.$ Choose $\mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$.

   For $\lambda = 3$: $(A - 3I)\mathbf{v} = \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = 0 \implies v_1 = v_2.$ Choose $\mathbf{v}_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.

3. Construct general solution: $\mathbf{X}(t) = C_1 e^{t}\begin{bmatrix} 1 \\ -1 \end{bmatrix} + C_2 e^{3t}\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$

4. Apply initial conditions (if provided) to find $C_1$ and $C_2$.

Would you like further clarification or an example with initial conditions solved?

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Related Questions:

1. How do you compute eigenvalues and eigenvectors for higher-dimensional matrices?
2. What happens when the eigenvalues are complex?
3. How can you use diagonalization to solve systems of differential equations?
4. Can you solve systems with non-constant coefficients using the matrix method?
5. How do repeated eigenvalues affect the solution of the system?

Tip:

For systems with complex eigenvalues, the solution will involve sine and cosine terms due to Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$.