1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.Is V a vector space? Why or why not?

Let, f and g be two functions in V. Then, f"(x) = f(x) - 5 g"(x) = g(x) - 5.

Answered: 1.T.2 Let V be the set of all…

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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1.T.1 Let U be the set of all twice-differentiable functions f : R → R satisfying f"(x)+ 2f'(x) – 3f (x) = 0. We want to show that U is a vector space. a) Let h(x) = f(x)+g(x) for two functions f(x) and g(x). Write down h"(x) and h'(x) in terms of f"(x) and g"(x) and f'(x) and g' (x). b) If f"(x)+2f'(x)-3f(x) = 0 and g" (x)+2g'(x)–3g(x) = 0, show that h" (x)+2h'(x)– 3h(x) = 0. c) Suppose that f(x) is in U, and cis a real number. Let k(x) = cf(x). Compute k"(x)+2k' (x)– 3k(x), and show that k(x) is in U. d) Verify that e² is in U. e) In class we verified that e-3z is in U. Is 5e² + 17e-3 in U? Why or why not?
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1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5. Is V a vector space? Why or why not?