Problem 32 - Differential Equations and Linear Algebra

Question

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

Step-by-Step Solution

Answer

The Wronskian of the functions \(f_1(x) = 1\), \(f_2(x) = x\), and \(f_3(x) = x^2\) is computed as $$ W(f_1(x), f_2(x), f_3(x)) = \begin{vmatrix} 1 & x & x^2 \\ 0 & 1 & 2x \\ 0 & 0 & 2 \end{vmatrix} = 2. $$ Since the Wronskian is a nonzero constant, the given functions are linearly independent on the interval \(I=(-\infty, \infty)\).

Step 1: Compute the derivatives

Before computing the Wronskian, we need the first and second derivatives of the functions \(f_{1}(x)\), \(f_{2}(x)\), and \(f_{3}(x)\). They are as follows: - \(f_{1}(x) = 1\): $$ f_{1}'(x) = 0, \quad f_{1}''(x) = 0. $$ - \(f_{2}(x) = x\): $$ f_{2}'(x) = 1, \quad f_{2}''(x) = 0. $$ - \(f_{3}(x) = x^{2}\): $$ f_{3}'(x) = 2x, \quad f_{3}''(x) = 2. $$

Step 2: Compute the Wronskian

Now that we have the function derivatives, we can compute the Wronskian using the given formula: $$ W(f_{1}(x), f_{2}(x), f_{3}(x)) = \begin{vmatrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ f_{1}'(x) & f_{2}'(x) & f_{3}'(x) \\ f_{1}''(x) & f_{2}''(x) & f_{3}''(x) \end{vmatrix} = \begin{vmatrix} 1 & x & x^{2} \\ 0 & 1 & 2x \\ 0 & 0 & 2 \end{vmatrix}. $$

Step 3: Evaluate the determinant

We can easily compute the determinant of the matrix by expanding along the first row. Doing so, we have: $$ W(f_{1}(x), f_{2}(x), f_{3}(x)) = \begin{vmatrix} 1 & x & x^{2} \\ 0 & 1 & 2x \\ 0 & 0 & 2 \end{vmatrix} = 1 \cdot \begin{vmatrix} 1 & 2x \\ 0 & 2 \end{vmatrix} = 1 \cdot (1\cdot2 - 0\cdot2x) = 2. $$

Step 4: Check if the Wronskian is nonzero

Since the Wronskian \(W(f_{1}(x), f_{2}(x), f_{3}(x)) = 2\) is a constant and nonzero, the given functions are linearly independent on the interval \(I=(-\infty, \infty)\).

Key Concepts

Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. They are central to understanding the natural world, as they describe phenomena from the microscopic behavior of particles to the motion of planets.

For instance, the functions provided in the exercise, when used in a differential equation, can describe a wide range of practical problems such as Newton's second law of motion, which can be represented by a second-order differential equation. Understanding whether solutions to these equations are unique and behave nicely often involves examining their linear independence through tools such as the Wronskian.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces and through matrices. It is a foundational topic that is used across sciences including physics, engineering, computer science, and economics to model and solve a wide array of problems. Matrices, which are rectangular arrays of numbers, are a central tool in linear algebra used to represent linear transformations, and the process of solving linear systems often involves operations with these matrices.

In the context of this exercise, linear algebra principles are used to calculate the Wronskian determinant, which is a matrix composed of the provided functions and their derivatives, to determine the linear independence of functions.

Determinants

The determinant is a scalar value that can be computed from the elements of a square matrix. It has important mathematical properties and is used in various areas including linear algebra, calculus, and more. Determinants help in solving linear equations, inverting matrices, and are also used in calculus to determine the volume scaling factor of linear transformations.

In our exercise, the determinant of the Wronskian matrix is evaluated to find out the linear independence of the functions. The fact that the determinant is nonzero provides us with valuable information: it confirms that our matrix and, therefore, the vector space generated by our functions, is non-degenerate, meaning that the functions are linearly independent.

Linear Independence

Linear independence is a key concept in linear algebra that determines whether a set of vectors (or functions in the space of solutions to a differential equation) can be expressed as linear combinations of each other. A set of functions is considered linearly independent if no function in the set can be written as a combination of the others. This is an important property when solving systems of equations or understanding the span of solutions to differential equations.

To illustrate, the functions in our exercise are said to be linearly independent if the Wronskian is nonzero for all x in the interval I, which is exactly what we've shown. This independence is crucial because it ensures that we have a complete basis for our solution space and can represent any solution to the associated differential equation using these functions.