Homework Help is Here – Start Your Trial Now!
Math
Calculus
We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y=0; x², x³, (0, 0) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₂ and c₂, not both zero, such that c₁x² + ₂x³ = 0. While this may be clear for these solutions that are different powers of x, we have a formal test to the linear independence. Recall the definition of the Wronskian for the case of two functions f₁ and f₂, each of which have a first derivative. w(f₂, f₂) = By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent. Let f₁(x) = x² and f₂(x)= x³. Complete the Wronskian for these functions. x² x3 w(x², x³) = 43 2x

We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y=0; x², x³, (0, 0) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₂ and c₂, not both zero, such that c₁x² + ₂x³ = 0. While this may be clear for these solutions that are different powers of x, we have a formal test to the linear independence. Recall the definition of the Wronskian for the case of two functions f₁ and f₂, each of which have a first derivative. w(f₂, f₂) = By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent. Let f₁(x) = x² and f₂(x)= x³. Complete the Wronskian for these functions. x² x3 w(x², x³) = 43 2x

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y = 0; x², x³, (0, ∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and C₂, not both zero, such that c₁x² + ₂x³ = 0. While this may be clear for these solutions that are different powers of x, we have a formal test to verify the linear independence. Recall the definition of the Wronskian for the case of two functions f₁ and f2, each of which have a first derivative. f₁ f₂ W(f₁, f₂) = f₁' f₂ By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent. Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. x² x³ 2-1 2x w(x², x³) =
Transcribed Image Text:We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y = 0; x², x³, (0, ∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and C₂, not both zero, such that c₁x² + ₂x³ = 0. While this may be clear for these solutions that are different powers of x, we have a formal test to verify the linear independence. Recall the definition of the Wronskian for the case of two functions f₁ and f2, each of which have a first derivative. f₁ f₂ W(f₁, f₂) = f₁' f₂ By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent. Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. x² x³ 2-1 2x w(x², x³) =
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS