Question

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(x)=x^{3}, \quad g(x)=|x|^{3}\)

Short answer

#Answer# The functions \(f(x)=x^3\) and \(g(x)=|x|^3\) are linearly dependent.

Step-by-step solution

Define the functions

We are given the following functions: \(f(x)=x^{3}\) \(g(x)=|x|^3\)

Compute the derivatives

In order to use the Wronskian, we need to find the first derivatives of our functions \(f(x)\) and \(g(x)\). Derivative of \(f(x)\): \(f'(x)=\frac{d}{dx}x^3=3x^2\) Derivative of \(g(x)\): For the derivative of \(g(x)=|x|^3\), we have two cases: 1. For \(x\geq0\), \(g(x)=x^3\), so \(g'(x)=3x^2\). 2. For \(x<0\), \(g(x)=(-x)^3=-x^3\), so \(g'(x)=-3x^2\). Thus, we have: $g'(x)=\begin{cases} 3x^2, & \text{if } x\geq 0 \\ -3x^2, & \text{if } x<0 \end{cases}$

Compute the Wronskian

Now, let's compute the Wronskian, defined as follows: $W(f,g)(x)=\begin{vmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{vmatrix}$ We plug in our functions and their derivatives: $W(f,g)(x)=\begin{vmatrix} x^3 & |x|^3 \\ 3x^2 & g'(x) \end{vmatrix}$ Now we need to find the determinant of this matrix. We need to consider the cases for \(x\geq0\) and \(x<0\) separately due to the piecewise function for \(g'(x)\).

Cases for the Wronskian

Let's compute the Wronskian for the two cases: Case 1: \(x\geq0\) $W(f,g)(x)=\begin{vmatrix} x^3 & x^3 \\ 3x^2 & 3x^2 \end{vmatrix}=(x^3)(3x^2)-(x^3)(3x^2)=0$ Case 2: \(x<0\) $W(f,g)(x)=\begin{vmatrix} x^3 & -x^3 \\ 3x^2 & -3x^2 \end{vmatrix}=(x^3)(-3x^2)-(x^3)(-3x^2)=0$ In both cases, the Wronskian is zero.

Conclusion

Since the Wronskian is zero for each of the cases, we can conclude that the functions \(f(x)=x^3\) and \(g(x)=|x|^3\) are linearly dependent.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They describe the relationship between a function and its rates of change. Often used in the fields of physics, engineering, biology, and economics, these equations help us understand how a particular system evolves over time.

In the context of our exercise, we are not directly solving a differential equation, but we're applying concepts that are central to the study of differential equations. Specifically, we're interested in determining whether two functions, which could be solutions to a differential equation, are linearly dependent or independent. This distinction is crucial because it can inform us about the uniqueness of solutions to differential equations and the behavior of systems modeled by these equations.

Linear Independence

Linear independence is a concept in vector spaces where vectors (or functions) are considered independent if no one of them can be expressed as a linear combination of the others. For a pair of functions, this means that there is no constant coefficient that can be attributed to one function to obtain the other.

In our example, we are examining two functions, f(x)=x^3 and g(x)=|x|^3, to see if they are linearly independent. The step-by-step solution involves checking if a constant multiple of one function can represent the other. Through the process of evaluating their Wronskian, we ultimately look for a non-zero determinant which would imply that the functions are independent. If the Wronskian is zero, as in our case, it suggests linear dependence.

Wronskian Determinant

The Wronskian determinant is a tool used in the analysis of differential equations to test for the linear independence of a set of functions. It is named after the Polish mathematician Józef Hoene-Wroński. The Wronskian involves calculating the determinant of a matrix constructed from the functions and their derivatives up to a certain order.

For the two functions given in our exercise, we calculated the Wronskian by creating a matrix where the first row is the functions themselves, and the second row is their first derivatives. We evaluated this determinant for different cases depending on the value of x. The Wronskian was zero in both cases we considered, which indicates that the functions are linearly dependent. Therefore, they cannot form a basis for the space of solutions to a related differential equation and are not uniquely determined by their initial conditions.