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Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. |x² w(x², x5)= 2x

Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. |x² w(x², x5)= 2x

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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We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 6xy' +10y = 0; x², x5, (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 + c₂x5 = 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 f₁₂ and f₂, each of which have a first derivative. ¹1 ¹2 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 +5 w(x², x5) W(f₁, f₂) = 2x =
Transcribed Image Text:We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 6xy' +10y = 0; x², x5, (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 + c₂x5 = 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 f₁₂ and f₂, each of which have a first derivative. ¹1 ¹2 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 +5 w(x², x5) W(f₁, f₂) = 2x =
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