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Tutorial Exercise Consider the differential equation x²6xy +10y=0; x², x5, (0,00). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x2y" 6xy+10y=0; x², x5, (0, 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 + c₂50. 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. w(f₂f₂) = 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 f(x)=x² and ₂(x)=x5. Complete the Wronskian for these functions. x² x5 w(x², x³) 2x

Tutorial Exercise Consider the differential equation x²6xy +10y=0; x², x5, (0,00). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x2y" 6xy+10y=0; x², x5, (0, 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 + c₂50. 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. w(f₂f₂) = 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 f(x)=x² and ₂(x)=x5. Complete the Wronskian for these functions. x² x5 w(x², x³) 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...
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Tutorial Exercise Consider the differential equation x²y" - 6xy' +10y = 0; x², x5, (0, ∞o). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 6xy' + 10y = 0; x², x5, (0, ∞o) 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 + ₂x5 = 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 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) = x5. Complete the Wronskian for these functions. x² x5 W(f₁, f₂) = w(x², x5)= 2x
Transcribed Image Text:Tutorial Exercise Consider the differential equation x²y" - 6xy' +10y = 0; x², x5, (0, ∞o). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 6xy' + 10y = 0; x², x5, (0, ∞o) 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 + ₂x5 = 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 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) = x5. Complete the Wronskian for these functions. x² x5 W(f₁, f₂) = w(x², x5)= 2x
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