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
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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