Problem 27 - Advanced Engineering Mathematics

Question

(a) The following commands can be used in Mathematica: \(\mathbf{A}=\\{\\{4,2\\},\\{3,3\\}\\}\) \(\mathbf{c}=\\{\mathbf{c} 1, \mathbf{c} 2\\}\) \(\mathrm{m}=\) MatrixExp \([\mathrm{A} \mathrm{t}]\) sol = Expand [m.c] Collect[sol, \(\\{\mathrm{c} 1, \mathrm{c} 2\\}] /\) /MatrixForm The output gives $$\begin{array}{l} x(t)=c_{1}\left(\frac{2}{5} e^{t}+\frac{3}{5} e^{6 t}\right)+c_{2}\left(-\frac{2}{5} e^{t}+\frac{2}{5} e^{6 t}\right) \\ y(t)=c_{1}\left(-\frac{3}{5} e^{t}+\frac{3}{5} e^{6 t}\right)+c_{2}\left(\frac{3}{5} e^{t}+\frac{2}{5} e^{6 t}\right). \end{array}$$ The eigenvalues are 1 and 6 with corresponding eigenvectors $$\left(\begin{array}{r} -2 \\ 3 \end{array}\right) \quad \text { and } \quad\left(\begin{array}{l} 1 \\ 1 \end{array}\right),$$ so the solution of the system is $$\mathbf{X}(t)=b_{1}\left(\begin{array}{r} -2 \\ 3 \end{array}\right) e^{t}+b_{2}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) e^{6 t}$$ or $$\begin{array}{l} x(t)=-2 b_{1} e^{t}+b_{2} e^{6 t} \\ y(t)=3 b_{1} e^{t}+b_{2} e^{6 t}. \end{array}$$ If we replace \(b_{1}\) with \(-\frac{1}{5} c_{1}+\frac{1}{5} c_{2}\) and \(b_{2}\) with \(\frac{3}{5} c_{1}+\frac{2}{5} c_{2},\) we obtain the solution found using the matrix exponential. (b) \(x(t)=c_{1} e^{-2 t} \cos t-\left(c_{1}+c_{2}\right) e^{-2 t} \sin t\) \(y(t)=c_{2} e^{-2 t} \cos t+\left(2 c_{1}+c_{2}\right) e^{-2 t} \sin t\)

Step-by-Step Solution

Answer

Express \(x(t)\) and \(y(t)\) using \(c_1, c_2\) with identified cosine and sine terms separately, verifying consistency.

Step 1: Initial Clarification

We need to solve part (b) of the exercise, which involves expressing the functions \(x(t)\) and \(y(t)\) as a combination of \(c_1\) and \(c_2\) in the given form of exponential and trigonometric functions.

Step 2: Break Down x(t)

The expression for \(x(t)\) is: \(x(t) = c_1 e^{-2t} \cos t - (c_1 + c_2) e^{-2t} \sin t\). We need to look at the components of cosine and sine separately and identify how they combine with the constants \(c_1\) and \(c_2\).

Step 3: Break Down y(t)

Similarly, the expression for \(y(t)\) is: \(y(t) = c_2 e^{-2t} \cos t + (2c_1 + c_2) e^{-2t} \sin t\). Again, separate the sine and cosine components to identify how \(c_1\) and \(c_2\) contribute.

Step 4: Combine Terms

Group terms of \(e^{-2t}\cos t\) and \(e^{-2t}\sin t\) for both \(x(t)\) and \(y(t)\). This helps in visually simplifying the expressions and confirms the linear combination of \(c_1\) and \(c_2\) for these trigonometric functions.

Step 5: Verify Expression Consistency

Check if the simplified expressions match any transformation of \(x(t)\) and \(y(t)\) when expressed in the given format. This also involves observing symmetry or patterns similar to the first part of the question where another function's structure was verified.

Key Concepts

Eigenvectors

An eigenvector is a special vector that, when a linear transformation is applied to it, does not change direction. Essentially, it only scales by a factor (which is the eigenvalue). Eigenvectors are deeply related to the structure of matrices and can provide insight into the geometry of linear transformations.

Let's use an example to clarify. Imagine a linear transformation represented by matrix \( \mathbf{A} \). When this transformation is applied to an eigenvector \( \mathbf{v} \), the result is a scaled version of \( \mathbf{v} \), but in the same direction.

For the matrix \( \mathbf{A} \) from the exercise, there are two eigenvectors given, \( \begin{pmatrix} -2 \ 3 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ 1 \end{pmatrix} \), which stay on their span under matrix \( \mathbf{A} \). Such vectors are crucial when determining the makeup and transformations described by differential equations.

Eigenvalues

Eigenvalues are the scalars associated with eigenvectors. When a matrix acts on its eigenvector, the eigenvalue is essentially the factor by which the eigenvector is stretched or compressed.

• In simpler terms, they tell you how a matrix modifies vectors on a particular line.
• In our problem, the matrix \( \mathbf{A} \) has eigenvalues 1 and 6.

This means when \( \mathbf{A} \) is multiplied by its eigenvectors, the result is a scaling of the vectors by a factor of either 1 or 6, respectively.

Eigenvalues can be found by solving the characteristic equation, which is derived by setting the determinant of \( ( \mathbf{A} - \lambda \mathbf{I} ) \) to zero, where \( \lambda \) represents the eigenvalues and \( \mathbf{I} \) the identity matrix. These values are vital in many applications like solving differential equations, as they significantly simplify the process by showing long-term behaviors in systems.

Differential Equations

Differential equations involve equations that contain derivatives, expressing how a particular function changes. They are vital in describing various real-world phenomena, such as motion, heat, and waves.

In the context of the exercise, these equations describe the rates of change of \( x(t) \) and \( y(t) \) over time.

• To solve them, we often use methods like eigenvectors and eigenvalues, as they can transform complex problems into manageable solutions.
• Here, the matrix exponential helps find a solution that describes how the state changes over time depending on initial conditions \( c_1 \) and \( c_2 \).

The system of differential equations from the exercise is solved via matrix exponentiation to yield expressions for \( x(t) \) and \( y(t) \) as functions of time, incorporating both exponential and sinusoidal components for a detailed behavior analysis.

Mathematica Programming

Mathematica is a powerful computational tool often used for solving complex mathematical problems, such as matrix exponentials and differential equations.

In the exercise, several Mathematica commands were used:

• **MatrixExp** to calculate the matrix exponential necessary for evolving systems over time.
• **Expand** and **Collect** to manipulate and simplify expressions, making them easier to understand and apply.

For students, this shows how computational software can be harnessed to solve mathematical problems effectively and verify manual solutions.

Mastering Mathematica not only helps with solving problems quickly but also with visualizing complex mathematical concepts, offering a more tangible understanding of abstract mathematical ideas.