Problem 39 - Differential Equations and Linear Algebra

Question

Show that the Wronskian of the given functions is identically zero on \((-\infty, \infty) .\) Determine whether the functions are linearly independent or linearly dependent on that interval. $$\begin{aligned} f_{1}(x)=2 x^{3}, \\ f_{2}(x) &=\left\\{\begin{aligned} 5 x^{3}, & \text { if } x \geq 0, \\ -3 x^{3}, & \text { if } x<0, \end{aligned}\right. \end{aligned}$$

Step-by-Step Solution

Answer

The Wronskian of the given functions \(f_{1}(x)=2 x^{3}\) and \(f_{2}(x)\), which has different definitions for \(x \geq 0\) and \(x < 0\), is found to be identically zero on the interval \((-\infty, \infty)\). This implies that the two functions are linearly dependent on that interval.

Step 1: STEP 1: Calculate the first derivatives of the functions

Let's first find the derivatives of the given functions. For \(f_1(x) = 2x^3\), its first derivative is: \(f_1'(x) = \frac{d}{dx}(2x^3) = 6x^2\) For \(f_2(x)\), we have a piecewise function: 1. If \(x \geq 0\), \(f_2(x) = 5x^3\). Therefore, its first derivative for this segment is: \(f_2'(x) = \frac{d}{dx}(5x^3) = 15x^2\) 2. If \(x < 0\), \(f_2(x) = -3x^3\). Therefore, its first derivative for this segment is: \(f_2'(x) = \frac{d}{dx}(-3x^3) = -9x^2\) Now that we have the first derivatives of both functions, we can move on to the next step.

Step 2: STEP 2: Calculate the Wronskian of the two functions

The Wronskian, denoted as \(W(f_1, f_2)\), is defined as: \( W(f_1, f_2) = \begin{vmatrix} f_1(x) & f_2(x) \\ f'_1(x) & f'_2(x) \end{vmatrix} \) We will first find \(W(f_1, f_2)\) for \(x \geq 0\), where \(f_2(x) = 5x^3\): \(W(f_1, f_2) = \begin{vmatrix} 2x^3 & 5x^3 \\ 6x^2 & 15x^2 \end{vmatrix} = (2x^3)(15x^2) - (5x^3)(6x^2) = 30x^5 - 30x^5 = 0 \) Now we will find \(W(f_1, f_2)\) for \(x < 0\), where \(f_2(x) = -3x^3\): \(W(f_1, f_2) = \begin{vmatrix} 2x^3 & -3x^3 \\ 6x^2 & -9x^2 \end{vmatrix} = (2x^3)(-9x^2) - (-3x^3)(6x^2) = -18x^5 + 18x^5 = 0 \) In both cases, the Wronskian is identically zero on the interval \((-\infty, \infty)\).

Step 3: STEP 3: Determine linear dependence or independence

As the Wronskian is identically zero on the entire interval \((-\infty, \infty)\), the two functions, \(f_1(x)\) and \(f_2(x)\), are linearly dependent on that interval.

Key Concepts

Linear Independence

Understanding linear independence is essential in algebra and calculus, especially when dealing with functions and vectors. Two functions (or vectors) are linearly independent if no scalar multiple of one function can be used to express the other. That means you cannot combine them to create the zero function unless all coefficients are zero.

In more practical terms, let's think about two vectors in a coordinate plane. If you cannot write one as a multiplication or a combination of the other, they are independent. In terms of functions, this means: if you have two functions, say \( f_1(x) \) and \( f_2(x) \), they are independent if there is no constant \( c \) such that \( c \cdot f_1(x) = f_2(x) \) for all \( x \).

Checking linear independence is crucial when solving differential equations, as it helps identify whether a solution set can span a function space fully. One common tool for assessing this is the Wronskian.

Linear Dependence

Linear dependence occurs when one function (or vector) can be composed of a linear combination of others. Simply put, you can express one function as a sum of scaled versions of the others.

For instance, let’s say you have two functions, \( f_1(x) \) and \( f_2(x) \). These functions are dependent if there exists a non-zero constant \( c \) such that \( f_2(x) = c \cdot f_1(x) \) everywhere on their domain.

An easy way to check for dependence, especially with functions, involves using the Wronskian determinant. If this determinant equals zero across an interval, it indicates that the functions are linearly dependent throughout that interval. This is what happened with \( f_1(x) = 2x^3 \) and \( f_2(x) \) in the problem.

Differential Equations

Differential equations are equations involving unknown functions and their derivatives. They are foundational because they describe various physical phenomena, such as heat transfer and motion.

Here are some key points about differential equations:

• They come in different forms: ordinary differential equations (ODEs) involve functions of a single variable, while partial differential equations (PDEs) involve multiple variables.
• Solving these requires finding a function (or set of functions) that satisfies the condition detailed in the equation.
• Solutions often involve determining linear independence or dependence between potential solutions.

The concept of linear independence in solutions ensures that the functions you've found cover all possibilities, without redundancy. This is where tools like the Wronskian come into play. They help determine if the solution set to a differential equation is valid and complete across its specified domain.