Explanation Given: The differential equation is y ′ ( t ) + 3 t y ( t ) = 1 − 2 t with the initial point y ( 2 ) = 0 . Calculation: The differential equation is y ′ ( t ) + 3 t y ( t ) = 1 − 2 t , where a ( t ) = 3 t and f ( t ) = 1 − 2 t . The integration factor of the differential equation y ′ ( t ) + 3 t y ( t ) = 1 − 2 t is computed as follows, p ( t ) = exp ( ∫ ( 3 t ) d t ) p ( t ) = exp ( 3 ln t ) p ( t ) = exp ( ln t 3 ) p ( t ) = t 3 Therefore, the required integrating factor of the differential equation y ′ ( t ) + 3 t y ( t ) = 1 − 2 t is p ( t ) = t 3 . Multiply the differential equation y ′ ( t ) + 3 t y ( t ) = 1 − 2 t by t 3 , t 3 y ′ ( t ) + t 3 × 3 t y ( t ) = t 3 ( 1 − 2 t ) t 3 y ′ ( t ) + 3 t 2 y ( t ) = t 3 − 2 t 4 t 3 y ′ ( t ) + 3 t 2 y ( t ) = t 3 − 2 t 4 Integrate on both sides with respect to t , ∫ t 3 y ′ ( t ) d t + ∫ 3 t 2 y ( t ) d t = ∫ ( t 3 − 2 t 4 ) d t Apply the Integration by parts and simplify further, ( t 3 ∫ y ′ ( t ) d t − ∫ ( ∫ y ′ ( t ) d t ) d ( t 3 ) d t d t ) + ∫ 3 t 2 y ( t ) d t = ∫ t 3

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter D1.4, Problem 46E

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To find: The initial value problem by computing the required integrating factor.

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Chapter D1.4, Problem 45E

Chapter D1.4, Problem 47E

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Calculus: Early Transcendentals (2nd Edition)

Ch. D1.1 - Prob. 1E Ch. D1.1 - Prob. 2E Ch. D1.1 - Prob. 3E Ch. D1.1 - If the general solution of a differential equation...Ch. D1.1 - Does the function y(t) = 2t satisfy the...Ch. D1.1 - Does the function y(t) = 6e3t satisfy the initial...Ch. D1.1 - Verifying general solutions Verify that the given...Ch. D1.1 - Verifying general solutions Verify that the given...Ch. D1.1 - Verifying general solutions Verify that the given...Ch. D1.1 - Verifying general solutions Verify that the given...

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The initial value problem by computing the required integrating factor.

Solution Summary: The author calculates the required integrating factor of the differential equation yprime.