Problem 16 - Advanced Engineering Mathematics

Question

In Problems 11-16, verify that the vector \(\mathbf{X}\) is a solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & 0 & 1 \\ 1 & 1 & 0 \\ -2 & 0 & -1 \end{array}\right) \mathbf{X} ; \quad \mathbf{X}=\left(\begin{array}{c} \sin t \\ -\frac{1}{2} \sin t-\frac{1}{2} \cos t \\ -\sin t+\cos t \end{array}\right) $$

Step-by-Step Solution

Answer

\( \mathbf{X} \) is a solution of the system.

Step 1: Differentiate X

Differentiate each component of the vector \( \mathbf{X} \) with respect to \( t \). The components of \( \mathbf{X} \) are: \( X_1 = \sin t \), \( X_2 = -\frac{1}{2} \sin t - \frac{1}{2} \cos t \), \( X_3 = -\sin t + \cos t \).\[ \mathbf{X}' = \begin{pmatrix} \cos t \ -\frac{1}{2} \cos t + \frac{1}{2} \sin t \ -\cos t - \sin t \end{pmatrix} \]

Step 2: Multiply Matrix with X

Multiply the matrix \( A \) by the vector \( \mathbf{X} \). The matrix \( A \) is:\[ A = \begin{pmatrix} 1 & 0 & 1 \ 1 & 1 & 0 \ -2 & 0 & -1 \end{pmatrix} \]Calculate \( A \mathbf{X} \):\[ A \mathbf{X} = \begin{pmatrix} 1 & 0 & 1 \ 1 & 1 & 0 \ -2 & 0 & -1 \end{pmatrix} \begin{pmatrix} \sin t \ -\frac{1}{2} \sin t - \frac{1}{2} \cos t \ -\sin t + \cos t \end{pmatrix} \]This results in:\[ A \mathbf{X} = \begin{pmatrix} \sin t + (-\sin t + \cos t) \ \sin t + \left(-\frac{1}{2} \sin t - \frac{1}{2} \cos t\right) \ -2\sin t - (-\sin t + \cos t) \end{pmatrix} \]Simplify each part:\[ = \begin{pmatrix} \cos t \ \frac{1}{2} \sin t - \frac{1}{2} \cos t \ -\cos t - \sin t \end{pmatrix} \]

Step 3: Verify Equality

Compare the result of \( A \mathbf{X} \) with \( \mathbf{X}' \):From Step 1:\[ \mathbf{X}' = \begin{pmatrix} \cos t \ -\frac{1}{2} \cos t + \frac{1}{2} \sin t \ -\cos t - \sin t \end{pmatrix} \]From Step 2:\[ A \mathbf{X} = \begin{pmatrix} \cos t \ \frac{1}{2} \sin t - \frac{1}{2} \cos t \ -\cos t - \sin t \end{pmatrix} \]Both these vectors are equal, therefore, \( \mathbf{X} \) is indeed a solution of the system.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation when working with linear systems. It's a way to apply transformations by mixing the rows and columns of matrices and vectors. In this exercise, to verify a solution, we multiply a matrix by a vector. The matrix, here, is a 3x3 matrix, and the vector is a 3x1 column vector.
When multiplying, you take each row of the matrix one at a time and perform a dot product with the vector:

• First row of the matrix is: \( \begin{pmatrix} 1 & 0 & 1 \end{pmatrix} \)
• The vector is: \( \begin{pmatrix} \sin t \ -\frac{1}{2} \sin t - \frac{1}{2} \cos t \ -\sin t + \cos t \end{pmatrix} \)
• The resulting product for this row and the vector is: \( \sin t + (- \sin t + \cos t) \).

Repeat similar steps with the other rows of the matrix. The resultant vector from this multiplication is compared with the derivative of the vector to check if they are equal. Matrix multiplication like this often serves to validate differential equation solutions.

Vector Functions

Vector functions are functions where each variable maps to a vector instead of a single number. Here, the vector function \( \mathbf{X} \) is defined in terms of trigonometric functions dependent on \( t \).
The importance of vector functions lies in their ability to describe dynamic systems:

• These functions can represent trajectories, forces, or any multi-variable data flow within a system.
• Components of the vector describe different states or conditions at any particular moment \( t \).

In the problem, each component such as \( \sin t \) provides an oscillatory behavior often used to model waves or periodic phenomena. Differentiating a vector function involves individually differentiating each component with respect to \( t \). This provides the rate of change for each element within the system, enabling analysis of how the entire system evolves over time.

System of Differential Equations

A system of differential equations is essentially a set of multiple differential equations that are connected. These typically represent a more complex system where multiple dependent variables affect each other.
In our exercise, we deal with three linked scalar equations bundled together in a matrix form:

• Each equation in the system could represent a different component of a physical system, like velocity, position, or acceleration.
• They involve first-order derivatives, indicating how each element in \( \mathbf{X} \) changes at every point \( t \).

The solution is shown to match the derivative, confirming it's a solution. Systems of differential equations often occur in modeling real-world phenomena, such as predator-prey models in ecology, the financial markets in economics, and mechanical systems in physics.