19 THE SET OF Funспопо f₁ (x) ex+2 LINEARLY DEPENDENT OR LINEARLY INDEPENDENT ON (-00,00) = AND F₂(x) = ex-3
19 THE SET OF Funспопо f₁ (x) ex+2 LINEARLY DEPENDENT OR LINEARLY INDEPENDENT ON (-00,00) = AND F₂(x) = ex-3
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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![**Problem Statement:**
Determine if the set of functions \( f_1(x) = e^{x+2} \) and \( f_2(x) = e^{x-3} \) are linearly dependent or linearly independent on the interval \((- \infty, \infty)\).
**Explanation:**
To assess the linear dependence or independence of these functions, one could use techniques such as the Wronskian or set up the equation for linear dependence:
\[ c_1 f_1(x) + c_2 f_2(x) = 0 \]
for all \( x \), where \( c_1 \) and \( c_2 \) are constants. If the only solution is \( c_1 = c_2 = 0 \), then the functions are linearly independent; otherwise, they are linearly dependent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fc33cc1-e822-44c8-a303-0b5fd2ff9f49%2F4a4d6f6a-e0c7-4ba2-912c-3c64773ca94c%2Frzua1to_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine if the set of functions \( f_1(x) = e^{x+2} \) and \( f_2(x) = e^{x-3} \) are linearly dependent or linearly independent on the interval \((- \infty, \infty)\).
**Explanation:**
To assess the linear dependence or independence of these functions, one could use techniques such as the Wronskian or set up the equation for linear dependence:
\[ c_1 f_1(x) + c_2 f_2(x) = 0 \]
for all \( x \), where \( c_1 \) and \( c_2 \) are constants. If the only solution is \( c_1 = c_2 = 0 \), then the functions are linearly independent; otherwise, they are linearly dependent.
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