Problem 10 - Advanced Engineering Mathematics

Question

In Problems 7-10, write the given system without the use of matrices. $$ \frac{d}{d t}\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{rr} 3 & -7 \\ 1 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)+\left(\begin{array}{l} 4 \\ 8 \end{array}\right) \sin t+\left(\begin{array}{c} t-4 \\ 2 t+1 \end{array}\right) e^{4 t} $$

Step-by-Step Solution

Answer

The system is: \( \frac{dx}{dt} = 3x - 7y + 4\sin t + (t-4)e^{4t} \) and \( \frac{dy}{dt} = x + y + 8\sin t + (2t+1)e^{4t} \).

Step 1: Understand the System Representation

The given system is written in matrix form, where the derivative vector \( \frac{d}{dt} \begin{pmatrix} x \ y \end{pmatrix} \) is expressed as the sum of the product of a matrix with vector \( \begin{pmatrix} x \ y \end{pmatrix} \), plus another vector expressed in terms of \( \sin t \) and \( e^{4t} \).

Step 2: Multiply the Matrix by the Vector

Compute the product of the matrix \( \begin{pmatrix} 3 & -7 \ 1 & 1 \end{pmatrix} \) with the vector \( \begin{pmatrix} x \ y \end{pmatrix} \) to distribute it across the two-dimensional system. This step transforms the matrix equation into a system of differential equations:\[ \frac{d}{dt}\begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 3x - 7y \ x + y \end{pmatrix} \]

Step 3: Incorporate Non-homogeneous Parts Involving \(\sin t\)

Add the vector involving \( \sin t \), \( \begin{pmatrix} 4 \ 8 \end{pmatrix} \sin t \), to each respective component in the vector from Step 2. This modifies the equations:\[ \begin{pmatrix} 3x - 7y + 4\sin t \ x + y + 8\sin t \end{pmatrix} \]

Step 4: Incorporate Non-homogeneous Parts Involving \(e^{4t}\)

Add the vector involving \( e^{4t} \), \( \begin{pmatrix} t-4 \ 2t+1 \end{pmatrix} e^{4t} \), to each respective component in the vector from Step 3. This results in the final system of equations:\[ \begin{pmatrix} 3x - 7y + 4\sin t + (t-4)e^{4t} \ x + y + 8\sin t + (2t+1)e^{4t} \end{pmatrix} \]

Step 5: Write the Separated Differential Equations

Express the entire system in the form of two individual differential equations:\( \frac{dx}{dt} = 3x - 7y + 4\sin t + (t-4)e^{4t} \) and \( \frac{dy}{dt} = x + y + 8\sin t + (2t+1)e^{4t} \).

Key Concepts

Matrix Representation

Matrix representation provides a compact and efficient way to express and analyze a system of linear differential equations. In essence, a matrix organizes the coefficients of the variables and functions involved in the system.
For the given problem, we started with a matrix that encapsulates the relationships between variables \( x \) and \( y \). It was given as follows:

• The matrix \( \begin{pmatrix} 3 & -7 \ 1 & 1 \end{pmatrix} \) represents the coefficients that act directly on the variables \( x \) and \( y \).
• This matrix is multiplied by the vector \( \begin{pmatrix} x \ y \end{pmatrix} \), indicating how each element interacts in the system.

By multiplying the matrix with the vector, we can derive individual equations for each component, which allows us to analyze the system dynamically. This representation is essential, as it structurally holds both coefficients and variables, making the manipulation of differential equations systematic and clear.

Non-Homogeneous Systems

A non-homogeneous system of differential equations includes terms independent of the system's variables, often incorporating functions such as \( \sin t \) and \( e^{4t} \) in this exercise.

The original equation incorporates non-homogeneous elements that complicate the initial straightforward matrix multiplication. For instance:

• The vector involving \( \sin t \), \( \begin{pmatrix} 4 \ 8 \end{pmatrix} \sin t \), is an external driving force or input to the system.
• The expression \( \begin{pmatrix} t-4 \ 2t+1 \end{pmatrix} e^{4t} \) is another addition that varies with time differently than the basic system produced by the matrix itself.

These non-homogeneous parts require additional handling in solving the differential equations, as they add complexity outside of what matrix multiplication alone covers. They are often approached by particular solution techniques, providing deeper insight into system behavior under varying conditions.

System of Equations

A system of equations involves multiple equations working together, often dependent on shared variables like \( x \) and \( y \) in this problem. Here, the system resulted in two differential equations that govern the dynamics of \( x \) and \( y \):

• The first equation is \( \frac{dx}{dt} = 3x - 7y + 4\sin t + (t-4)e^{4t} \).
• The second is \( \frac{dy}{dt} = x + y + 8\sin t + (2t+1)e^{4t} \).

By expressing our matrix expression into these two separate equations, we simplify the task of solving the system.

Each equation reveals how changes in time affect \( x \) and \( y \), and how these variables, in turn, interact with each other. This isolation into two parts facilitates finding solutions either analytically or numerically, further helped by initial or boundary conditions. Understanding and manipulating such systems are key in applications across physics, engineering, and applied mathematics.