Problem 10 - Advanced Engineering Mathematics

Question

In Problems 1-12, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right) X $$

Step-by-Step Solution

Answer

General solution: \( X(t) = C_1 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} + C_2 e^{2t} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + C_3 e^{2t} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \).

Step 1: Identify the Matrix A

The system of differential equations is given by \( X' = AX \), where \( A \) is the coefficient matrix. From the problem, identify \( A = \begin{pmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{pmatrix} \). Our goal is to find the general solution of this system.

Step 2: Find Eigenvalues of A

The eigenvalues \( \lambda \) of the matrix \( A \) are found by solving the characteristic equation \( \det(A - \lambda I) = 0 \). The matrix \( A - \lambda I \) is \( \begin{pmatrix} 1-\lambda & 0 & 1 \ 0 & 1-\lambda & 0 \ 1 & 0 & 1-\lambda \end{pmatrix} \). The determinant of this matrix is \( (1-\lambda) \left[(1-\lambda)^2 - 1\right] \). Expanding, we get the characteristic equation \( (1-\lambda)((1-\lambda)^2 - 1) = 0 \). Simplify to find the eigenvalues: \( \lambda_1 = 0 \) (multiplicity 1), and \( \lambda_2 = 2 \) (multiplicity 2).

Step 3: Find Eigenvectors for \( \lambda = 0 \)

Substitute \( \lambda = 0 \) into \( A - \lambda I \) to find the eigenvector(s). This gives us \( \begin{pmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{pmatrix} - \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{pmatrix} \). Solve \( AX = 0 \) for \( X \). The system becomes \( \begin{pmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \). The solutions are spanned by the vector \( \begin{pmatrix} -1 \ 0 \ 1 \end{pmatrix} \).

Step 4: Find Eigenvectors for \( \lambda = 2 \)

Substitute \( \lambda = 2 \) into \( A - \lambda I \) and solve \( (A - 2I)X = 0 \). The matrix \( A - 2I \) is \( \begin{pmatrix} -1 & 0 & 1 \ 0 & -1 & 0 \ 1 & 0 & -1 \end{pmatrix} \). The solutions to this system are spanned by \( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix} \).

Step 5: Construct the General Solution

Using the eigenvalues and corresponding eigenvectors, construct the general solution. The solution is a linear combination of terms of the form \( C_i e^{\lambda_i t} v_i \), where \( C_i \) are constants, \( \lambda_i \) are eigenvalues, and \( v_i \) are eigenvectors. The general solution is: \[ X(t) = C_1 e^{0t} \begin{pmatrix} -1 \ 0 \ 1 \end{pmatrix} + C_2 e^{2t} \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} + C_3 e^{2t} \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}. \]This simplifies to:\[X(t) = C_1 \begin{pmatrix} -1 \ 0 \ 1 \end{pmatrix} + C_2 e^{2t} \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} + C_3 e^{2t} \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}. \]

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are essential concepts in the study of linear algebra and systems of differential equations. Imagine you have a matrix, which acts as an operator that can stretch or compress vectors in different directions.
An eigenvalue (\(\lambda\)) is a scalar that gives you the factor by which the eigenvector is stretched when the matrix is applied. To find eigenvalues, you solve the characteristic equation \(\det(A - \lambda I) = 0\), where \(A\) is your matrix and \(I\) is the identity matrix of similar dimensions.

Eigenvectors are the non-zero vectors that only get scaled by that factor, not redirected, under the matrix transformation. Each eigenvalue can have one or more corresponding eigenvectors.

• Characteristic Equation: Derived from \(\det(A - \lambda I) = 0\), it helps to find eigenvalues.
• Eigenvalues (\(\lambda\)): Values that satisfy the characteristic equation.
• Eigenvectors: Vectors that, when multiplied by the matrix, yield a scalar multiple of the original vector.

Learning how to compute eigenvalues and eigenvectors is crucial as they help to simplify complex matrix operations and are key in finding the general solution for systems of differential equations.

Matrix Algebra

Matrix algebra provides the necessary operations and techniques for manipulating matrices and solving various mathematical problems including systems of linear equations and differential equations.
Understanding the basic operations like addition, multiplication, and finding the inverse is crucial.
In this context, matrices can be used to represent linear transformations, enabling us to perform operations that satisfy certain properties.

• Matrix Multiplication: Involves dot product of rows and columns and it's crucial in solving systems of equations.
• Determinant: A special number assigned to a square matrix, useful in solving systems of linear equations and finding eigenvalues.
• Identity Matrix: Acts as a multiplicative identity for matrices, similar to the number 1 in arithmetic.
• Inverse: The matrix that, when multiplied with the original matrix, yields the identity matrix.

Matrix algebra is indispensable in solving complex problems and in theoretical explorations across physics, engineering, computer sciences, and more.

Systems of Linear Equations

Systems of linear equations are collections of one or more linear equations involving the same set of variables. They form the backbone of mathematical models that interpret various phenomena in science and engineering.
In this context, a system can be neatly represented in matrix form which paves the way for using algebraic methods to find solutions.
For example, using matrices, we can solve the system by transforming it to a row-echelon form or using methods such as Gaussian elimination.

• Representation: Systems can be represented as \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the vector of variables, and \(B\) is the constants vector.
• Solutions: Can be unique, infinite, or nonexistent given the rank of the matrix and consistency of the system.
• Matrix Methods: Techniques like Gaussian elimination and Cramer's Rule are typical solutions strategies.

Mastering systems of linear equations is vital as they are the gateway to understanding more complex phenomena through mathematical modeling.