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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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Certainly! Here's a transcription suitable for an educational website: --- **9. Use the information that the functions 1, x, x², ..., xⁿ are linearly independent on the real line and the definition of linear independence to prove directly that, for any constant r, the functions f₀(x) = eʳˣ, f₁(x) = xeʳˣ, f₂(x) = x²eʳˣ, ..., fₙ(x) = xⁿeʳˣ are linearly independent on the whole real line.** If the functions f₁, f₂, f₃, ..., fₙ are linearly independent on the real line, then the identity c₁f₁(x) + c₂f₂(x) + ... + cₙfₙ(x) = 0 holds on I only when (1) _________. To prove that the functions f₀(x) = eʳˣ, f₁(x) = xeʳˣ, f₂(x) = x²eʳˣ, ..., fₙ(x) = xⁿeʳˣ are linearly independent, the corresponding identity is (2) _________. *What is the next step to prove the given functions are linearly independent?* A. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, xeʳˣ. B. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, xⁿeʳˣ. C. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, x²eʳˣ. D. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, eʳˣ. Apply the above property to the identity. Thus, __________ = 0. Since the product of two terms is zero, either (3) _________. Can eʳˣ be zero? ☐ No ☐ Yes Thus,
Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website: --- **9. Use the information that the functions 1, x, x², ..., xⁿ are linearly independent on the real line and the definition of linear independence to prove directly that, for any constant r, the functions f₀(x) = eʳˣ, f₁(x) = xeʳˣ, f₂(x) = x²eʳˣ, ..., fₙ(x) = xⁿeʳˣ are linearly independent on the whole real line.** If the functions f₁, f₂, f₃, ..., fₙ are linearly independent on the real line, then the identity c₁f₁(x) + c₂f₂(x) + ... + cₙfₙ(x) = 0 holds on I only when (1) _________. To prove that the functions f₀(x) = eʳˣ, f₁(x) = xeʳˣ, f₂(x) = x²eʳˣ, ..., fₙ(x) = xⁿeʳˣ are linearly independent, the corresponding identity is (2) _________. *What is the next step to prove the given functions are linearly independent?* A. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, xeʳˣ. B. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, xⁿeʳˣ. C. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, x²eʳˣ. D. ☐ Write the above sum in terms of the linear independent functions 1, x, x², ..., xⁿ by isolating the common factor, eʳˣ. Apply the above property to the identity. Thus, __________ = 0. Since the product of two terms is zero, either (3) _________. Can eʳˣ be zero? ☐ No ☐ Yes Thus,
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