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
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...
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