Explanation Consider the polar coordinates x = r cos θ , y = r sin θ . Differentiate the polar coordinates with respect to t and simplify as follows. d x d t = − r sin θ d θ d t + cos θ d r d t = y + x − x ( x 2 + y 2 ) = r sin θ + r cos θ − r 3 cos θ ...... ( 1 ) Similarly, for the y coordinate differentiate as follows. d y d t = r cos θ d θ d t + sin θ d r d t = − x + y − y ( x 2 + y 2 ) = − r cos θ + r sin θ − r 3 sin θ ...... ( 2 ) To solve for d r d t add equation 1 and 2 after multiplying equation 1 with cos θ and equation 2 with sin θ . cos 2 θ d r d t + sin 2 θ d r d t = cos θ ( r sin θ + r cos θ − r 3 cos θ ) + sin θ ( − r cos θ + r sin θ − r 3 sin θ ) d r d t = r cos θ sin θ + r cos 2 θ − r 3 cos 2 θ − r cos θ sin θ + r sin <

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 16.5, Problem 2E

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To show: That any trajectory starting on the unit circle x2+y2=1 will traverse the unit circle in a periodic solution.

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Chapter 16.5, Problem 1E

Chapter 16.5, Problem 3E

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Chapter 16 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 1–4, match the differential equations...Ch. 16.1 - In Exercises 5 and 6, copy the slope fields, and...Ch. 16.1 - In Exercises 5 and 6, copy the slope fields, and...Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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That any trajectory starting on the unit circle x 2 + y 2 = 1 will traverse the unit circle in a periodic solution.

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To show: That any trajectory starting on the unit circle x2+y2=1 will traverse the unit circle in a periodic solution.

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Chapter 16 Solutions