Prove that if F(2) is a function with a finite number of linearly independent derivatives, i.e., if F(n)(x), F(n-1)(x),..., F'(x), F(x) are linearly independ- ent functions, where n is a finite number, then F(z) consists only of such terms as a, x, ea, sin ax, cos ax, and combinations of such terms, where a is a constant and k is a positive integer. Hint. Set the linear combination of these functions equal to zero, i.e., set C₂F(n)(x) + C-1F(n-¹)(x) +...+C₁F'(x) + CoF(x) = 0, where the C's are not all zero, and then show, by Lesson 20, that the only functions F(x) that can satisfy this equation are those stated.
Prove that if F(2) is a function with a finite number of linearly independent derivatives, i.e., if F(n)(x), F(n-1)(x),..., F'(x), F(x) are linearly independ- ent functions, where n is a finite number, then F(z) consists only of such terms as a, x, ea, sin ax, cos ax, and combinations of such terms, where a is a constant and k is a positive integer. Hint. Set the linear combination of these functions equal to zero, i.e., set C₂F(n)(x) + C-1F(n-¹)(x) +...+C₁F'(x) + CoF(x) = 0, where the C's are not all zero, and then show, by Lesson 20, that the only functions F(x) that can satisfy this equation are those stated.
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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Transcribed Image Text:2. Prove that if F(x) is a function with a finite number of linearly independent
derivatives, i.e., if F(n)(x), F(n-1)(x), ···, F¹(x), F(x) are linearly independ-
ent functions, where n is a finite number, then F(x) consists only of such terms
as a, x, ea, sin ax, cos ax, and combinations of such terms, where a is a
constant and k is a positive integer. Hint. Set the linear combination of these
functions equal to zero, i.e., set
C₂F(n)(x) + C₁-1F(n-¹)(x) +...+C₁F'(x) + CoF(x) = 0,
where the C's are not all zero, and then show, by Lesson 20, that the only
functions F(x) that can satisfy this equation are those stated.
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