Explanation Property (a) of the given theorem states that the product of two even functions is even. Consider two functions f and g both are even. The product of functions is given below. h ( x ) = f ( x ) g ( x ) Replace − x for x in the above equation. h ( − x ) = f ( − x ) g ( − x ) = f ( x ) g ( x ) = h ( x ) So, the product of two even functions is even. Therefore, property (a) of given theorem is verified. Now, Property (c) of the given theorem states that the product of an even function and an odd function is odd. Consider the function f is even and function g is odd. The product of functions is given below. h ( x ) = f ( x ) g ( x ) Replace − x for x in the above equation. h ( − x ) = f ( − x ) g ( − x ) = f ( x ) ( − g ( x ) ) = − h ( x ) Therefore, the product of an even function and an odd function is odd. Hence, the property (c) of given theorem is verified. Now, Property (d) of the given theorem states that the sum (difference) of two even functions is even. Consider function f and g both are even. The sum (difference) of functions is given below. h ( x ) = f ( x ) ± g ( x ) Replace − x for x in the above equation. h ( − x ) = f ( − x ) ± g ( − x ) = f ( x ) ± g ( x ) = h ( x ) Therefore, the sum (difference) of two even functions is even. Hence, the property (d) of given theorem is verified. Now, Property (f) of the given theorem states that when function f is even, then ∫ − a a f ( x ) d x = 2 ∫ 0 a f ( x ) d x . Consider function f is even. ∫ − a a f ( x ) d x = ∫ − a 0 f ( x ) d x + ∫ 0 a f ( x ) d x Assume, u = − x , then d u = − d x . Substitute u for − x and − d u for d x in the above equation

Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term

9th Edition

ISBN: 9781337604901

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 11.3, Problem 52E

To determine

The verification of the properties (a), (c), (d), (f), and (g) from the given theorem.

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Chapter 11 Solutions

Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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