Question

In Exercises \(1-16\) find the general solution. $$ \mathbf{y}^{\prime}=\frac{1}{3}\left[\begin{array}{rr} 7 & -5 \\ 2 & 5 \end{array}\right] \mathbf{y} $$

Short answer

Answer: The general solution to the given matrix differential equation is $\mathbf{y}(t) = e^{3t}\begin{bmatrix} \frac{5}{2}c_1 + \frac{1}{2}c_2 \\ c_1 + \frac{1}{2}c_2 \end{bmatrix}$.

Step-by-step solution

Write the Given Matrix A

We can write down the given matrix \(A\) as follows: $$ A = \frac{1}{3}\left[\begin{array}{rr} 7 & -5 \\ 2 & 5 \end{array}\right] $$

Find the Eigenvalues of Matrix A

The eigenvalues of matrix \(A\) can be found by solving the characteristic equation, which is given by: $$ \det{(A-\lambda I)} = 0, $$ where \(I\) is the identity matrix of size \(2\times2\). Then, we have: $$ \det{\left(\frac{1}{3}\left[\begin{array}{rr} 7-\lambda & -5 \\ 2 & 5-\lambda \end{array}\right]\right)} = 0. $$ Expanding the determinant, we obtain: $$ \frac{1}{9}\left((7-\lambda)(5-\lambda) - (-5)(2)\right) = 0. $$ Solving for \(\lambda\), we find the eigenvalues: $$ \lambda_1 = 3, \quad \lambda_2 = 3. $$

Find the Eigenvectors for Each Eigenvalue

For each eigenvalue, we will find the eigenvectors by solving the following equation: $$ (A-\lambda_i I) \mathbf{v}_i = 0, $$ where \(\mathbf{v}_i\) is the eigenvector corresponding to the eigenvalue \(\lambda_i\). Since both eigenvalues are the same, we will find only one linearly independent eigenvector. Plugging in \(\lambda_1 = 3\) into the equation, we have: $$ \frac{1}{3}\left[\begin{array}{rr} 4 & -5 \\ 2 & 2 \end{array}\right] \mathbf{v}_1 = 0. $$ By solving the linear system, we find one linearly independent eigenvector: $$ \mathbf{v}_1 = \begin{bmatrix} 5/2 \\ 1 \end{bmatrix}. $$

Construct the Fundamental Matrix

Since we have only one linearly independent eigenvector, we need to construct a fundamental matrix using generalized eigenvectors. To find a generalized eigenvector, we now solve the equation: $$ (A-\lambda_1 I)\mathbf{w} = \mathbf{v}_1, $$ where \(\mathbf{w}\) is the generalized eigenvector. Plugging in the values, we get: $$ \frac{1}{3}\left[\begin{array}{rr} 4 & -5 \\ 2 & 2 \end{array}\right] \mathbf{w} = \begin{bmatrix} 5/2 \\ 1 \end{bmatrix}. $$ Solving this linear system, we find the generalized eigenvector: $$ \mathbf{w} = \begin{bmatrix} 1/2 \\ 1/2 \end{bmatrix}. $$ Now, we can construct the fundamental matrix \(X\) as follows: $$ X(t) = \begin{bmatrix} \mathbf{v}_1(t) & \mathbf{w}(t)e^{\lambda_1 t} \end{bmatrix} = \begin{bmatrix} 5/2 & 1/2 \\ 1 & 1/2 \end{bmatrix}e^{3t}. $$

Compute the General Solution

Using the fundamental matrix \(X(t)\), we can write the general solution of the given matrix differential equation as follows: $$ \mathbf{y}(t) = X(t) \mathbf{c} = \begin{bmatrix} 5/2 & 1/2 \\ 1 & 1/2 \end{bmatrix}e^{3t} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}, $$ where $\mathbf{c} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}$ is an arbitrary constant vector. So, the general solution to the given matrix differential equation is: $$ \mathbf{y}(t) = e^{3t}\begin{bmatrix} \frac{5}{2}c_1 + \frac{1}{2}c_2 \\ c_1 + \frac{1}{2}c_2 \end{bmatrix}. $$

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is essential when working with matrices, especially in the context of differential equations. An eigenvalue is a special scalar that results from multiplying a matrix by a vector, which is called an eigenvector. The equation that ties these together is \( A\mathbf{v} = \lambda \mathbf{v} \), where \( A \) is a matrix, \( \lambda \) is the eigenvalue, and \( \mathbf{v} \) is the eigenvector.
To find eigenvalues, you solve the characteristic equation: \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix. This equation provides the values of \( \lambda \) that make the determinant zero.

• Repeated Eigenvalues: Sometimes, matrices have repeated eigenvalues, which can involve special considerations when finding eigenvectors.
• Finding Eigenvectors: For each eigenvalue, eigenvectors are found by solving \( (A - \lambda I) \mathbf{v} = 0 \). Sometimes more than one linearly independent eigenvector can exist for a single eigenvalue.

This knowledge is fundamental for analyzing and solving matrix differential equations.

Matrix Differential Equation

A matrix differential equation is an equation in which the unknown quantity is a vector-valued function, and its derivative is expressed in terms of a matrix. These types of equations are written in the form \( \mathbf{y}' = A\mathbf{y} \), where \( \mathbf{y} \) is the vector of unknown functions and \( A \) is a constant matrix.
Solving matrix differential equations involves finding a function \( \mathbf{y}(t) \) that satisfies the equation for all points \( t \). In practice, this often involves finding eigenvalues and eigenvectors of the matrix \( A \).
These solutions help in:

• Representing complex systems such as population models, mechanical structures, or electrical circuits.
• Allowing for the expression of solutions using a matrix exponential, especially for linear systems.

Matrix differential equations are a compact and powerful way to model and solve real-world problems.

General Solution

The general solution of a differential equation refers to the complete set of solutions that can be constructed with arbitrary constants, which capture all possible initial conditions. For matrix differential equations, the general solution is often derived using the fundamental matrix.
Here's how it works:

• You use the fundamental matrix \( X(t) \) constructed from the eigenvectors and, if necessary, generalized eigenvectors.
• The general solution is then expressed as \( \mathbf{y}(t) = X(t) \mathbf{c} \), where \( \mathbf{c} \) is the vector of constants determined by initial conditions.

This method transforms the problem of solving differential equations into one of finding general solutions using linear algebra techniques. It's crucial for understanding the ultimate behavior and responses of the systems represented by the equations.

Fundamental Matrix

The fundamental matrix is a key concept when it comes to solving linear systems of differential equations. It is essentially a matrix whose columns are constructed from the solutions (eigenvectors and possibly generalized eigenvectors) of a differential system.
A fundamental matrix \( X(t) \) is often represented as:

• A collection of solutions, systematically organized into an array.
• It's key that \( X(t) \) satisfies \( X'(t) = AX(t) \), where \( A \) is the coefficient matrix from the differential equation \( \mathbf{y}' = A\mathbf{y} \).

The construction involves:

• Using eigenvectors for distinct eigenvalues.
• In the case of repeated eigenvalues, employing generalized eigenvectors to ensure the matrix is complete.

Understanding the fundamental matrix helps in piecing together the general solution for the system of differential equations. This concept allows us to formally and structuredly tackle complex systems.