Problem 33 - A First Course in Differential Equations with Modeling Applications

Question

Find linearly independent functions that are annihilated by the given differential operator. $$D^{3}-10 D^{2}+25 D$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given a third-order linear differential operator \(D^3 - 10D^2 + 25D\). Our task is to find linearly independent functions that are annihilated by this operator, meaning they satisfy the equation \((D^3 - 10D^2 + 25D)y = 0\).

Step 2: Factor the Operator

To solve the differential equation, factor the differential operator. The operator \(D^3 - 10D^2 + 25D\) can be factored as \(D(D-5)^2\), because \(D^3 - 10D^2 + 25D = D(D-5)^2\).

Step 3: Solve Subproblems Using Roots

Now solve each part of the factored operator separately. The equations are: 1. \(Dy = 0\) implying \(y = C_1\), where \(C_1\) is a constant.2. \((D-5)y = 0\) implying \(y = C_2e^{5x}\), where \(C_2\) is a constant.3. \((D-5)^2y = 0\) implying \(y = (C_3 + C_4x)e^{5x}\), where \(C_3\) and \(C_4\) are constants.

Step 4: Identify Linearly Independent Solutions

From the subproblems, we derive the solutions that are linearly independent: \(1\), \(e^{5x}\), and \(xe^{5x}\). These solutions form the set of linearly independent functions.

Key Concepts

Linearly Independent Functions

In mathematics, and particularly in the study of differential equations, determining linearly independent functions is crucial. These functions are significant because they form the basis solutions to a differential equation, especially when the equation is homogeneous.
Linearly independent functions are those that cannot be written as a linear combination of each other. In simpler terms, no function in the set can be expressed as a sum of multiples of the other functions. This property makes them quite valuable in forming general solutions to differential equations.

• For example, in the case provided, the solutions we determined were: \(1\), \(e^{5x}\), and \(xe^{5x}\).
• These functions are linearly independent because no one of these can be expressed as a combination of the others.

In the context of this exercise, each solution satisfies the differential equation individually, which highlights their independence.

Differential Operator

A differential operator is a tool used to describe differentiation concerning a function. It's commonly denoted by \(D\), which represents the derivative taken with respect to the independent variable, most often \(x\).
Operators can be combined to form more complex expressions, as seen in the problem statement: \(D^3 - 10D^2 + 25D\). This operator is a polynomial expression of derivatives.

• Such an operator can act on a function to produce another function, effectively changing its behavior.
• In solving differential equations, recognizing and understanding how differential operators work is essential.

They are the cornerstones for setting up equations in classical mechanics, quantum mechanics, and many areas of engineering and physics.

Factorization of Operators

Factorization is a powerful technique applied not only in algebra but also in differential equations to simplify and solve them. By breaking down a differential operator into simpler components, we can solve complex equations in parts.
In the exercise, the operator \(D^3 - 10D^2 + 25D\) is factored into \(D(D-5)^2\). This factorization transforms a single complex problem into a series of simpler problems.

• Each factor relates to a simpler differential operation that can be solved independently.
• This step-by-step approach aids in finding all solutions to the differential equation.

This method significantly enriches one's toolbox for solving higher-order differential equations by reducing their complexity.