Program Explanation Given information: The differential equation is y ″ + y = 3 cos 2 x and the function is y 1 = cos x − cos 2 x and y 2 = sin x − cos 2 x . Calculation: Label the given function as y 1 = cos x − cos 2 x ........ (1) Label the given function as y 2 = sin x − cos 2 x ........ (2) Label the given differential equation as y ″ + y = 3 cos 2 x ........ (3) Differentiate equation (1) with respect to x as shown below. y ′ 1 = d d x ( cos x − cos 2 x ) = − sin x + 2 sin 2 x Again differentiate y ′ 1 = − sin x + 2 sin 2 x with respect to x as y ″ 1 = d d x ( − sin x + 2 sin 2 x ) = − cos x + 2 ( 2 ) cos 2 x = − cos x + 4 cos 2 x Substitute − cos x + 4 cos 2 x for y ″ and cos x − cos 2 x for y in the left-hand side ( L .H .S ) of the equation (3) as L .H .S = − cos x + 4 cos 2 x + cos x − cos 2 x = 4 cos 2 x − cos 2 x = 3 cos 2 x The right-hand side ( R .H .S ) of the equation (3) is 3 cos 2 x . Hence, the left-hand side and right-hand side of the differential equation is the same. Hence, by substitution, the function y 1 = cos x − cos 2 x satisfies the given differential equation y ″ + y = 3 cos 2 x

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ISBN: 9780135904206

Author: Edwards

Publisher: PEARSON

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Chapter 1.1, Problem 8P

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To show: That the given function is a solution of the given differential equation by substitution.

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Chapter 1.1, Problem 7P

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Chapter 1 Solutions

DIFFERENTIAL EQUATIONS+..-MYLAB ACCESS

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That the given function is a solution of the given differential equation by substitution.

Solution Summary: The author explains that the given function solves the differential equation by substitution. The equation is y′′+y=3mathrmcos2x

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To show: That the given function is a solution of the given differential equation by substitution.

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Chapter 1 Solutions