Problem 34 - A First Course in Differential Equations with Modeling Applications

Question

Find linearly independent functions that are annihilated by the given differential operator. $$D^{2}(D-5)(D-7)$$

Step-by-Step Solution

Answer

The linearly independent functions are 1, \( x \), \( e^{5x} \), and \( e^{7x} \).

Step 1: Understand the Differential Operator

The given differential operator is \( D^2(D-5)(D-7) \). Each factor of the operator implies a root or solution for the differential equation. The operator acts on a function \( y \) such that when applied to \( y \), the result is zero.

Step 2: Identify Roots from the Operator

The roots can be determined by setting each factor to zero. From \( D^2 \), we have a double root at 0, and from \( D-5 \) and \( D-7 \), we have simple roots at 5 and 7, respectively. This yields the roots \( r = 0, 5, 7 \).

Step 3: Construct Solutions from Roots

Based on the roots, the general solution consists of linearly independent functions. For the double root at 0, the solutions are \( e^{0x} = 1 \) and \( xe^{0x} = x \). For the simple roots at 5 and 7, the solutions are \( e^{5x} \) and \( e^{7x} \).

Step 4: Verify Linearly Independent Solutions

The functions 1, \( x \), \( e^{5x} \), and \( e^{7x} \) are linearly independent because they correspond to different exponential and polynomial forms. Each function cannot be expressed as a linear combination of the others.

Key Concepts

Linearly Independent Functions

In mathematics, linearly independent functions are crucial to understanding solutions of differential equations. Two or more functions are said to be linearly independent if no function can be written as a linear combination of the others. This means combining the functions with constants, other than zero, doesn't construct another function in the group. To determine if a set of functions is linearly independent, consider the following criteria:

• There should be at least two functions.
• None of the functions can be expressed as a combination of others.

Suppose we have functions 1, 𝑥, 𝑒^{5x}, and 𝑒^{7x}, derived from the differential operator. For linear independence: - There is no constant multiplication of 1 that results in 𝑥. - Similarly, no combination of 𝑥, 𝑒^{5x}, or 𝑒^{7x} forms any of the others. This quality of being different in expression ensures that their combination forms all possible solutions without redundancy.

Annihilator Method

The annihilator method is a technique in differential equations, especially useful for finding a particular solution of non-homogeneous linear differential equations. Here, the differential operator acts like a filter to eliminate certain functions by making their image zero. To use the annihilator method:

• Identify a differential operator that cancels out the unwanted function.
• Apply this operator to both sides of the equation.

In the given exercise, the operator \(D^2(D-5)(D-7)\) is applied to seek functions that transform to zero. This results in polynomial and exponential functions that the operator can 'annihilate': Functions like 1, \(x\), \(e^{5x}\), and \(e^{7x}\) satisfy this condition as they are roots of the operator.

Roots of Differential Equations

The concept of roots is central to solving differential equations. The roots of a differential equation derived from a characteristic polynomial dictate the solutions' nature. Each root corresponds to a function in the general solution. To find these roots: 1. Start with the differential operator, represented as a product of terms. Each term corresponds to a root. 2. Set each term to zero to find its root value. For example, from \(D^2(D-5)(D-7)\), we derive roots as follows:

• \(D^2\) gives a double root at zero, indicating solutions like 1 and \(x\).
• \(D-5\) and \(D-7\) give simple roots at 5 and 7 respectively, leading to solutions \(e^{5x}\) and \(e^{7x}\).

These roots help in constructing the solution space while ensuring each function derived fits into the general solution, contributing to its completeness.