Explanation Given information: The functions f ( x ) and g ( x ) satisfy the following differential equation. f ″ ( x ) = − f ( x ) ; g ″ ( x ) = − g ( x ) And f ( 0 ) = 0 ; f ′ ( 0 ) = 1 ; g ( 0 ) = 1 ; g ′ ( 0 ) = 0 Formula used: d d x ( sin x ) = cos x d d x ( cos x ) = − sin x Calculation: Now let us consider the function as shown below. f ( x ) = sin x Differentiate the above function with respect to x . f ′ ( x ) = cos x Differentiate the above function with respect to x again. f ′ ′ ( x ) = − sin x = − f ( x ) So, f ( x ) = sin x satisfies the equation f ″ ( x ) = − f ( x ) . Now let us consider the function as shown below. g ( x ) = cos x Differentiate the above function with respect to x . g ′ ( x ) = − sin x Differentiate the above function with respect to x again. g ″ ( x ) = − cos x = − g ( x ) So, g ( x ) = cos x satisfies the equation g ″ ( x ) = − g ( x ) . So, f ( x ) = sin x and g ( x ) = cos x are the solution of f ″ ( x ) = − f ( x ) and g ″ ( x ) = − g ( x ) respectively. Now we will prove that the solutions are unique. Consider the expression as shown below. f ( x ) − g ( x ) = h ( x ) And [ h ( x ) ] 2 + [ h ′ ( x ) ] 2 Differentiate the above function with respect to x . d d x ( [ h ( x ) ] 2 + [ h ′ ( x ) ] 2 ) = d d x ( [ h ( x ) ] 2 ) + d d x ( [ h ′ ( x ) ] 2 ) = 2 h ( x ) h ′ ( x ) + 2 h ′ ( x ) h ″ ( x ) = 2

4th Edition

ISBN: 9781319309671

Author: Rogawski

Publisher: MAC HIGHER

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Limits

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Question

Chapter 4.3, Problem 81E

To determine

To prove:

That f(x)=sinx and g(x)=cosx is unique solution of the differential equation f″(x)=−f(x) and g″(x)=−g(x) for all x.

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Chapter 4.3, Problem 80E

Chapter 4.4, Problem 1PQ

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Chapter 4 Solutions

CALCULUS: EARLY TRANS 4TH ED W/ ACCESS

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To prove: That f ( x ) = sin x and g ( x ) = cos x is unique solution of the differential equation f ″ ( x ) = − f ( x ) and g ″ ( x ) = − g ( x ) for all x .

Solution Summary: The author demonstrates that f(x)=mathrmsinx is unique solution of the differential equation.

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To prove:

That f(x)=sinx and g(x)=cosx is unique solution of the differential equation f″(x)=−f(x) and g″(x)=−g(x) for all x.

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Chapter 4 Solutions

To prove: That f ( x ) = sin x and g ( x ) = cos x is unique solution of the differential equation f ″ ( x ) = − f ( x ) and g ″ ( x ) = − g ( x ) for all x .

Solution Summary: The author demonstrates that f(x)=mathrmsinx is unique solution of the differential equation.

Concept explainers

Videos

To prove: That f(x)=sinx and g(x)=cosx is unique solution of the differential equation f″(x)=−f(x) and g″(x)=−g(x) for all x.

Want to see the full answer?

Chapter 4 Solutions

To prove:

That f(x)=sinx and g(x)=cosx is unique solution of the differential equation f″(x)=−f(x) and g″(x)=−g(x) for all x.