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Consider the differential equation x²y" - 4xy' + 6y = 0; Verify that the given functions form Form the general solution. x², x³, (0, ∞). a fundamental set of solutions of the differential equation on the indicated interval.

Consider the differential equation x²y" - 4xy' + 6y = 0; Verify that the given functions form Form the general solution. x², x³, (0, ∞). a fundamental set of solutions of the differential equation on the indicated interval.

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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Consider the differential equation x²y" - 4xy' + 6y=0; x², x³, (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" - 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 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 w(f₁, f₂) = w(x², x³) = 2x
Transcribed Image Text:Consider the differential equation x²y" - 4xy' + 6y=0; x², x³, (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" - 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 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 w(f₁, f₂) = w(x², x³) = 2x
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