Question

Determine whether the given functions form a fundamental set of solutions for the linear system. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right] \mathbf{y}, \quad \mathbf{y}_{1}(t)=\left[\begin{array}{c} e^{t} \\ -e^{t} \end{array}\right], \quad \mathbf{y}_{2}(t)=\left[\begin{array}{l} e^{-t} \\ e^{-t} \end{array}\right] $$

Short answer

Answer: Yes, the given functions form a fundamental set of solutions because their Wronskian is nonzero.

Step-by-step solution

Calculate the Wronskian of the given solutions

The Wronskian of the given functions \(\mathbf{y}_{1}(t)\) and \(\mathbf{y}_{2}(t)\) is defined as: $$W(\mathbf{y}_{1}(t),\mathbf{y}_{2}(t))=\begin{vmatrix} \mathbf{y}_{1,1}(t) & \mathbf{y}_{2,1}(t)\\ \mathbf{y}_{1,2}(t) & \mathbf{y}_{2,2}(t) \end{vmatrix}$$ We plug in the given functions and obtain: $$W(\mathbf{y}_{1}(t),\mathbf{y}_{2}(t))=\begin{vmatrix} e^{t} & e^{-t}\\ -e^{t} & e^{-t} \end{vmatrix}$$

Check if the Wronskian is nonzero

Now, we calculate the determinant of the matrix: $$W(\mathbf{y}_{1}(t),\mathbf{y}_{2}(t))=(e^{t})(e^{-t})-(-e^{t})(e^{-t})$$ $$W(\mathbf{y}_{1}(t),\mathbf{y}_{2}(t))=e^{t-t}+e^{t-t}=1+1=2$$ Since the Wronskian is nonzero, the given functions \(\mathbf{y}_{1}(t)\) and \(\mathbf{y}_{2}(t)\) form a fundamental set of solutions for the linear system.

Key concepts

Fundamental Set of Solutions

In mathematics, when dealing with systems of differential equations, identifying a fundamental set of solutions is crucial. Consider a homogeneous linear system. A set of solutions is deemed fundamental if they are linearly independent and span the solution space of the differential equation. In simpler terms, these solutions cover all possibilities for that system.

For the provided problem, the given functions have been deemed to form a fundamental set. Why? Because their Wronskian is nonzero. This non-zero value tells us that the solutions are linearly independent. Essentially, this means there are no overlaps or redundancies in the forms of the solutions. Hence, they sufficiently capture the dynamics of the entire system.

In practice, a fundamental set of solutions helps solve initial value problems and other related tasks where the full behavior of a system is required.

Wronskian

The Wronskian is a mathematical tool used to determine whether a set of solutions to a differential equation is linearly independent. Think of it as a litmus test for independence. For a set of functions, if their Wronskian is non-zero at some point, those functions are linearly independent over an interval.

To calculate the Wronskian for the functions given, you set up a matrix with each function and its derivative as columns. You then compute the determinant of this matrix. For instance, in our problem:

• Column 1: The first function and its derivative.
• Column 2: The second function and its derivative.

In our specific problem, after computation, this determinant equals 2, which is very much not zero! This confirms that our solutions, \(\mathbf{y}_{1}(t)\) and \(\mathbf{y}_{2}(t)\), are indeed linearly independent. And thus, a fitting fundamental set of solutions.

Linear System

When exploring differential equations, particularly those structured in matrices, you come across linear systems quite often. A linear system in differential equations involves expressions in the form of matrix multiplication of vectors and matrices. It's linear because its behavior is described by first-degree expressions (those that are not squared or multiplied by themselves).

The original exercise presents a linear system depicted as \( \mathbf{y}^{\prime} = A\mathbf{y} \) where \( A \) is a constant matrix, in this case, \[\begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix}\]
Such systems are pivotal in many fields, like physics and engineering, due to their predictability and ease when seeking solutions. Each solution of these systems, such as \(\mathbf{y}_{1}(t)\) and \(\mathbf{y}_{2}(t)\), is to be plugged back into the system to verify accuracy.

Understanding how linear systems operate is fundamental for mathematical modeling, providing insights into how entities interact under specified rules.