Problem 11 - Advanced Engineering Mathematics

Question

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} -1 & -1 & 0 \\ \frac{3}{4} & -\frac{3}{2} & 3 \\ \frac{1}{8} & \frac{1}{4} & -\frac{1}{2} \end{array}\right) $$

Step-by-Step Solution

Answer

Find eigenvalues, solve for eigenvectors, express as linear combination.

Step 1: Write Down the System

We are given the system \( \mathbf{X}^{\prime} = A \mathbf{X} \), where \( A \) is the matrix \[ A = \begin{pmatrix} -1 & -1 & 0 \ \frac{3}{4} & -\frac{3}{2} & 3 \ \frac{1}{8} & \frac{1}{4} & -\frac{1}{2} \end{pmatrix}. \] Our task is to find the general solution of this system.

Step 2: Find the Eigenvalues

To find the eigenvalues of the matrix \( A \), we solve the characteristic equation \( \det(A - \lambda I) = 0 \). First, calculate \( A - \lambda I \): \[ \begin{pmatrix} -1-\lambda & -1 & 0 \ \frac{3}{4} & -\frac{3}{2}-\lambda & 3 \ \frac{1}{8} & \frac{1}{4} & -\frac{1}{2}-\lambda \end{pmatrix}. \] Find the determinant and solve for \( \lambda \).

Step 3: Calculate the Determinant

Expand the determinant for \( \det(A - \lambda I) \): \[ \det\begin{pmatrix} -1-\lambda & -1 & 0 \ \frac{3}{4} & -\frac{3}{2}-\lambda & 3 \ \frac{1}{8} & \frac{1}{4} & -\frac{1}{2}-\lambda \end{pmatrix} = (-1-\lambda)\left((-\frac{3}{2}-\lambda)(-\frac{1}{2}-\lambda) - \frac{3}{4}\right). \] Simplify and solve the resulting polynomial equation to find the eigenvalues.

Step 4: Solve for Eigenvectors

For each eigenvalue \( \lambda \) found, solve \( (A - \lambda I)\mathbf{v} = 0 \) to find the corresponding eigenvectors. Substitute each eigenvalue into \( A - \lambda I \) and use row reduction to find the non-trivial solutions of the homogeneous system, giving the eigenvectors.

Step 5: Write the General Solution

The general solution of \( \mathbf{X}^{\prime} = A \mathbf{X} \) is given by \( \mathbf{X}(t) = c_1 e^{\lambda_1 t} \mathbf{v_1} + c_2 e^{\lambda_2 t} \mathbf{v_2} + c_3 e^{\lambda_3 t} \mathbf{v_3} \), where \( \lambda_i \) are the eigenvalues and \( \mathbf{v_i} \) are the corresponding eigenvectors (after finding and substituting specific expressions.)

Step 6: Verify and Simplify

Verify each solution by substituting into the original differential system equation to ensure correctness. Simplify to ensure that all parts are consistent with the found general solution.

Key Concepts

Eigenvalues

In any system of differential equations, eigenvalues play a crucial role. They help us understand how different components of the system change over time. To find the eigenvalues of a matrix, which represents the system, you need to solve the characteristic equation given by \( \det(A - \lambda I) = 0 \). Here, \( A \) is the matrix, \( \lambda \) represents the eigenvalues we are solving for, and \( I \) is the identity matrix.

This involves:

• Subtracting \( \lambda \) from every diagonal element of the matrix \( A \).
• Calculating the determinant of this new matrix.
• Solving the resulting polynomial equation to find the values of \( \lambda \).

Once you find the eigenvalues, you can tell if a system is stable or unstable based on whether these values are positive, negative, or complex numbers.

Eigenvectors

Once you figure out the eigenvalues, the next step is to determine the corresponding eigenvectors. Eigenvectors provide a direction in which the system evolves. To find them, substitute an eigenvalue \( \lambda \) back into the matrix equation \((A - \lambda I)\mathbf{v} = 0\).

Then, you:

• Replace each eigenvalue into the matrix \( A \), reducing it through matrix subtraction.
• Solve for vector \( \mathbf{v} \) using techniques like row reduction to achieve a row-echelon form.

This process gives multiple solutions, which are scaled versions of each other, delineating the potential directions of system solutions. Finding eigenvectors for each eigenvalue allows us to build a full picture of the solutions to the system of differential equations.

Matrix Theory

Matrix theory underpins the process of solving systems of differential equations. A matrix is a powerful tool that encapsulates an entire linear system. With matrix operations, we can manipulate and solve these systems efficiently. In our exercise, we use matrix theory to simplify and solve the system \( \mathbf{X}^{\prime} = A \mathbf{X} \), where \( A \) is our central matrix.

Key concepts:

• **Matrices**: Arrays of numbers representing coefficients in systems of equations.
• **Determinants**: A scalar value offering insights into the matrix’s properties, used to find eigenvalues.
• **Identity Matrix**: Used during matrix subtraction when computing eigenvalues.

Understanding these concepts ensures you can apply matrix operations effectively, helping solve and interpret systems of differential equations.

System of Differential Equations

A system of differential equations consists of multiple equations that describe how variables change concerning each other over time. These systems can model a wide array of real-world scenarios, from ecological systems to electrical circuits. To solve such systems, we often transform them into a matrix form as drawn in our original exercise: \( \mathbf{X}^{\prime} = A \mathbf{X} \).

Steps to solve include:

• **Convert into Matrix Form**: Representing coefficients and variables as a matrix.
• **Determine Eigenvalues and Eigenvectors**: Predict system behavior and provide solution structures.
• **Construct the General Solution**: Using the eigenvalues and eigenvectors to express general solutions as combinations of these basic solutions.

By mastering these steps, the solutions to complex systems become more approachable and understandable, opening avenues to deeper analysis and application.