Problem 34 - Differential Equations and Linear Algebra

Question

Use the Wronskian to show that the given functions are linearly independent on the given interval \(I\). $$\begin{array}{l} f_{1}(x)=1, f_{2}(x)=3 x, f_{3}(x)=x^{2}-1, \\ I=(-\infty, \infty). \end{array}$$

Step-by-Step Solution

Answer

The Wronskian determinant for the functions \(f_1(x)=1, f_2(x)=3x, f_3(x)=x^2-1\) is found to be \(W(x)=6x\). As this determinant is non-zero over the given interval \(I=(-\infty, \infty)\), we can conclude that the given functions are linearly independent on this interval.

Step 1: Find the first derivatives of all given functions

First, we compute the first derivative of each function with respect to x: 1. \(f_1(x) = 1\): The derivative is \(f_1'(x) = 0\). 2. \(f_2(x) = 3x\): The derivative is \(f_2'(x) = 3\). 3. \(f_3(x) = x^2 - 1\): The derivative is \(f_3'(x)= 2x\).

Step 2: Set up the Wronskian determinant

The Wronskian determinant W(x) is defined as $$ W(x)=\underbrace{\left|\begin{array}{ccc} \hline 1 & 3x & x^2 - 1\\ 0 & 3 & 2x\\ \end{array}\right|}_{\text{Wronskian determinant}} $$

Step 3: Evaluate the determinant

To find the Wronskian determinant, we evaluate the 3 by 2 determinant \[ W(x)= \begin{vmatrix} 1 & 3x & x^2 - 1\\ 0 & 3 & 2x\\ \end{vmatrix} = 1 \cdot \begin{vmatrix} 3 & 2x\\ \end{vmatrix} - 3x \cdot \begin{vmatrix} 0 & 2x\\ \end{vmatrix} + (x^2 - 1) \cdot \begin{vmatrix} 0 & 3\\ \end{vmatrix} \] Let's compute each product separately: 1. \(1 \cdot \begin{vmatrix}3 & 2x\end{vmatrix} = 1 \cdot (3(2x)) = 6x\) 2. \(3x \cdot \begin{vmatrix}0 & 2x\end{vmatrix} = 3x \cdot 0 = 0\) 3. \((x^2 - 1) \cdot \begin{vmatrix}0 & 3\end{vmatrix} = (x^2 - 1) \cdot 0 = 0\) Now, plug the results back into the Wronskian: \[ W(x) = 6x \]

Step 4: Check if the determinant is non-zero on the interval

Now, we have to check whether the Wronskian determinant \(W(x) = 6x\) is non-zero over the interval \(I=(-\infty, \infty)\). Since 6x can be any nonzero value for values of x in the interval, it is non-zero over the given interval. Therefore, the given functions \(f_1(x)=1, f_2(x)=3x, f_3(x)=x^2-1\) are linearly independent on the interval \((-∞,∞)\).

Key Concepts

Linear Independence

In mathematics, understanding whether a set of functions is linearly independent is crucial, especially when dealing with solutions of differential equations.
Two or more functions are considered linearly independent if no function in the set can be written as a linear combination of the others.
This means coefficients that multiply these functions to make them equal to zero must all be zero.

• For example, consider functions \(f_1(x), f_2(x),\) and \(f_3(x)\).
• If \(c_1f_1(x) + c_2f_2(x) + c_3f_3(x) = 0\) only satisfies when \(c_1, c_2, c_3\) are all zero, these functions are declared linearly independent.

Linear independence is a fundamental concept. It helps us understand solution spaces of linear differential equations and confirms whether solutions are genuinely distinct and necessary.

Differential Equations

Differential equations play a significant role in modeling and solving various physical and mathematical problems. They involve equations that consist of an unknown function and its derivatives. Here, we look at how differential equations relate to the concept of the Wronskian determinant.
The solutions to a differential equation can be expressed as functions. Determining whether these functions are linearly independent can heavily influence the general solution of such an equation.

• The Wronskian is an essential tool for testing the linear independence of solutions to differential equations.
• By evaluating the Wronskian, one can determine the uniqueness and existence of solutions.

Thus, differential equations and their solutions form a cornerstone in the study of dynamical systems in sciences and engineering, where the independence of solutions is crucial for precise modeling.

Determinants

Determinants are a mathematical expression calculated from a square matrix, and they are crucial when working with systems of linear equations.
When we use a Wronskian to determine the linear independence of functions, we calculate the determinant of a matrix formed by these functions and their derivatives.

• For example, a central part of analyzing linear independence is constructing the Wronskian determinant matrix.
• This involves utilizing both the original functions and their derivatives.

If the determinant is non-zero in an interval, the functions are linearly independent in that interval.
This property makes the determinant calculation a core step in using Wronskians, offering a concise solution method for certainty in function independence.

First Derivatives

First derivatives provide the primary rate of change of a function, forming the essential basis for constructing the Wronskian.
Given a set of functions, computing their first derivatives is the initial step in many analysis procedures, including determining their linear independence using the Wronskian.

• Each function in the set is differentiated, and these derivatives form part of the matrix used in the Wronskian determinant.
• Derivatives transform the original function, detailing how they change concerning their input.

This transformation through differentiation creates a new layer of analysis, giving insight into the dynamic behavior of functions.
In this way, first derivatives are a foundational tool in both calculus and the study of differential equations.