Problem 32 Find a fundamental set of soluti... [FREE SOLUTION]

91影视

Elementary Differential Equations with Boundary Value Problems

William F. Trench, PhD

$Math Studyset 91影视 Explanations$ Math

2013 Edition

Chapter 9: Problem 32

Find a fundamental set of solutions. $$ (D-2)^{3}(D+1)^{2} D y=0 $$

Short Answer

Expert verified

Answer: The fundamental set of solutions for the given linear differential equation consists of the following functions: $y_1(x) = e^{2x}$, $y_2(x) = xe^{2x}$, $y_3(x) = x^2 e^{2x}$, $y_4(x) = e^{-x}$, and $y_5(x) = x e^{-x}$.

Step by step solution

Analyze the given equation

The given linear differential equation is: $$ (D-2)^{3}(D+1)^{2} D y=0 $$ The equation contains repeated roots, specifically the roots 2 and -1, each with multiple orders (3 for 2 and 2 for -1).

Write the general form of the solution for repeated roots

For the given equation, the general form of the solution is: $$ y(x) = C_1 e^{2x} + C_2 xe^{2x} + C_3 x^2 e^{2x} + C_4 e^{-x} + C_5 x e^{-x} $$ Where $C_1$, $C_2$, $C_3$, $C_4$, and $C_5$ are constants.

Identify a fundamental set of solutions

From the general form of the solution, the fundamental set of solutions consists of: 1. $y_1(x) = e^{2x}$ 2. $y_2(x) = xe^{2x}$ 3. $y_3(x) = x^2 e^{2x}$ 4. $y_4(x) = e^{-x}$ 5. $y_5(x) = x e^{-x}$ These five functions form a fundamental set of solutions to the given linear differential equation.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Differential Equation

A linear differential equation is one in which the dependent variable and all its derivatives appear to the first power and are not multiplied together. This type of equation is essential because it allows for methods that can systematically find solutions.
Generally, a linear differential equation can be written in the form:

$ L[y] = 0 $

where $ L $ is a linear operator, involving differential operators like $ D = \frac{d}{dx} $. Understanding the structure of these equations is vital to finding their solutions.

Fundamental Set of Solutions

A fundamental set of solutions refers to a group of solutions to a differential equation that can be used to express any solution to the equation. For a linear differential equation with constant coefficients, the general solution can often be written as a linear combination of a set of independent solutions.
The independence of these solutions is crucial because they form a basis for the solution space. To verify if a set is fundamental, you check if the Wronskian determinant of these solutions is non-zero.
In the given exercise, the functions:

$ y_1(x) = e^{2x} $
$ y_2(x) = xe^{2x} $
$ y_3(x) = x^2 e^{2x} $
$ y_4(x) = e^{-x} $
$ y_5(x) = x e^{-x} $

form the fundamental set of solutions.

Repeated Roots

Repeated roots occur when solving the characteristic equation of a differential equation results in some roots appearing more than once. For example, in the given problem, the roots 2 and -1 each have multiplicities of 3 and 2, respectively.
This happens often with higher-order differential equations and affects the form of the solution.
For a root $ r $ repeated $ n $ times, each solution involves increasing powers of $ x $:

$ e^{rx} $
$ xe^{rx} $
... until
$ x^{n-1} e^{rx} $

This pattern ensures all solutions remain linearly independent.

Solution Form

The solution form of a differential equation can vary depending on the nature of its roots. With repeated roots, as seen in the exercise, the solution adopts a form that accommodates these repetitions by including powers of $ x $ multiplied by the exponential term related to the root.
This leads to solutions such as:

$ C_1 e^{2x} + C_2 xe^{2x} + C_3 x^2e^{2x} $ for the root 2
$ C_4 e^{-x} + C_5 x e^{-x} $ for the root -1

By understanding the basic form, you can construct a comprehensive solution that includes all possible cases.
Subsequently, any solution to the differential equation is a combination of elements of this form.

Linear Operator Notation

Linear operator notation simplifies the way we express and manipulate differential equations. Here, the operator $ D $ denotes differentiation with respect to $ x $, such that $ Dy = \frac{dy}{dx} $.
In linear differential equations, operations are often compactly represented using these operators, like in our problem:

$ (D-2)^3(D+1)^2 D y = 0 $

This notation allows us to focus on the roots and their multiplicities instead of cumbersome manipulation of derivatives.
It is especially helpful in breaking down and solving complex, higher-order differential equations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find a fundamental set of solutions. $$ (D-2)^{3}(D+1)^{2} D y=0 $$

Short Answer

Step by step solution

Analyze the given equation

Write the general form of the solution for repeated roots

Identify a fundamental set of solutions

Key Concepts

Linear Differential Equation

Fundamental Set of Solutions

Repeated Roots

Solution Form

Linear Operator Notation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Logic and Functions

Decision Maths

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.