Problem 17 - Advanced Engineering Mathematics

Question

The given vectors are solutions of a system \(\mathbf{X}^{\prime}=\mathbf{A X}\). Determine whether the vectors form a fundamental set on the interval \((-\infty, \infty)\). $$ \mathbf{X}_{1}=\left(\begin{array}{l} 1 \\ 1 \end{array}\right) e^{-2 t}, \mathbf{X}_{2}=\left(\begin{array}{r} 1 \\ -1 \end{array}\right) e^{-6 t} $$

Step-by-Step Solution

Answer

The vectors form a fundamental set on \((- ibf, \infty)\) because the Wronskian is non-zero for all \(t\).

Step 1: Write the Vectors in Matrix Form

Write the given vectors as a matrix: \(\mathbf{X} = \begin{pmatrix} \mathbf{X}_1 & \mathbf{X}_2 \end{pmatrix} = \begin{pmatrix} e^{-2t} & e^{-6t} \ e^{-2t} & -e^{-6t} \end{pmatrix}\).

Step 2: Calculate the Wronskian

The Wronskian \(W(t)\) is given by the determinant of \(\mathbf{X}\):\[ W(t) = \begin{vmatrix} e^{-2t} & e^{-6t} \ e^{-2t} & -e^{-6t} \end{vmatrix} = e^{-2t}(-e^{-6t}) - e^{-6t}(e^{-2t}) \].

Step 3: Simplify the Wronskian

Simplify the expression:\[ W(t) = -e^{-8t} - e^{-8t} = -2e^{-8t} \].

Step 4: Determine if the Wronskian is Zero

Since \(W(t) = -2e^{-8t}\) is never zero for all real \(t\), the vectors form a fundamental set on \((-ibf, \infty)\).

Key Concepts

Wronskian Determinant

The Wronskian determinant is a powerful tool in the study of differential equations. It helps us determine whether a set of solutions to a linear differential equation is linearly independent, which is crucial for checking if these solutions form a fundamental set. For two functions, say \( f(t) \) and \( g(t) \), their Wronskian \( W(t) \) is calculated by finding the determinant:

• \( W(t) = \begin{vmatrix} f(t) & g(t) \ f'(t) & g'(t) \end{vmatrix} = f(t)g'(t) - g(t)f'(t) \)

In the context of systems of equations, we extend this concept to vectors. For the vector solutions given in the exercise, the Wronskian is calculated in a similar way but using the matrix formed by these vectors. It determines the linear independence and thus tells us if they cover enough solution space to be a fundamental set.

Linear Differential Equations

Linear differential equations are equations involving an unknown function and its derivatives. They can be thought of as equations that have solutions forming vector spaces. Such equations can typically be written in the form:

• \(a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \cdots + a_1(t)y' + a_0(t)y = g(t)\)

Where \(a_n(t), a_{n-1}(t), \ldots, a_0(t)\) are coefficients and \(g(t)\) is a known function. When dealing with systems of these equations, like in this exercise, we encounter matrix forms. Such a system is defined by a matrix equation \( \mathbf{X}' = \mathbf{A} \mathbf{X} \), representing a vector of unknown functions and their derivatives that need to be solved together.

Vector Solutions

A vector solution to a system of differential equations represents a combination of functions that solve the equation together. Think of each vector as a stack of functions neatly lined up, each element representing a different component of the system. In our exercise, each vector \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \) corresponds to solutions of the system, formed by expressions such as \( \begin{pmatrix} 1 \ 1 \end{pmatrix} e^{-2t} \). The concept of vector solutions is handy, as it transforms a complex system of equations into a manageable matrix equation. This especially helps when looking for a fundamental set of solutions, where finding linearly independent vector solutions ensures that every possible solution of the system can be considered a combination of these fundamental vectors.

System of Equations

Systems of equations are collections of multiple equations that relate several variables. In the context of differential equations, these systems describe how changes in some quantities are interconnected. Such a system might be expressed by \( \mathbf{X}' = \mathbf{A} \mathbf{X} \), indicating that the rate of change of a vector quantity \( \mathbf{X} \) depends on itself through the transformation matrix \( \mathbf{A} \). Solving these systems often requires determining a set of solutions that are fundamental, meaning they span the solution space. Linear independence of these vector solutions is then verified using methods like calculating the Wronskian. When working with such systems, visualizing them as connected paths where variables affect each other can help in grasping the underlying structure and solutions.