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Step 1 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 verify that the solutions are linearly independent. That is, there do not exist constants c, 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 f, and f₂, each of which have a first derivative. w(f₂f₂) = 4 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 f(x)=x² and f₂(x)=x². Complete the Wronskian for these functions. x³ w(x², x³)= 2x Step 2 Find the determinant. - w(x³, x²)= 3x² - (x²)(3x²) - (2x)(x³) The Wronskian [is not 3x² ✓equal to 0 for every x in the interval (0, 0), therefore the set of solutions ---Select--linearly independent. Gelect are not

Step 1 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 verify that the solutions are linearly independent. That is, there do not exist constants c, 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 f, and f₂, each of which have a first derivative. w(f₂f₂) = 4 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 f(x)=x² and f₂(x)=x². Complete the Wronskian for these functions. x³ w(x², x³)= 2x Step 2 Find the determinant. - w(x³, x²)= 3x² - (x²)(3x²) - (2x)(x³) The Wronskian [is not 3x² ✓equal to 0 for every x in the interval (0, 0), therefore the set of solutions ---Select--linearly independent. Gelect are not

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y = 0; x², x³, (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 + 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. x³ w(x², x³) = Step 2 2x Find the determinant. w(x³, x²) = 3x² 2x 3x² = (x²)(3x²) - (2x)(x³) The Wronskian is not 3.x² ✓equal to 0 for every x in the interval (0, ∞o), therefore the set of solutions ---Select--- linearly independent. ---Select- are are not
Transcribed Image Text:Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y = 0; x², x³, (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 + 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. x³ w(x², x³) = Step 2 2x Find the determinant. w(x³, x²) = 3x² 2x 3x² = (x²)(3x²) - (2x)(x³) The Wronskian is not 3.x² ✓equal to 0 for every x in the interval (0, ∞o), therefore the set of solutions ---Select--- linearly independent. ---Select- are are not
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