Explanation Given: The differential equation, d y d t = t 2 3 + 5 t . Formula used: Integration by parts, ∫ u d v = u v − ∫ v d u . Calculation: The given differential equation is written as, d y d t = t 2 3 + 5 t d y = t 2 3 + 5 t d t Integration both sides, ∫ d y = ∫ t 2 3 + 5 t d t y = ∫ t 2 3 + 5 t d t + C Now, let I = ∫ t 2 3 + 5 t d t . To integrate I by parts method can be used by taking u = t 2 and d v = d t 3 + 5 t . Differentiating u and integrating the 2 nd one we get, d u = 2 t d t And v = ( 1 5 ) ( 3 + 5 t ) 1 2 1 2 = 2 5 ( 3 + 5 t ) 1 2 Hence by using by parts method the original can be written as, ∫ u d v = u v − ∫ v d u = t 2 { 2 5 ( 3 + 5 t ) 1 2 } − ∫ { 2 5 ( 3 + 5 t ) 1 2 } ( 2 t d t ) = 2 t 2 5 ( 3 + 5 t ) 1 2 − 4 5 ∫ t ( 3 + 5 t ) 1 2 d t Again, for the second part by take u = t and d v = ( 3 + 5 t ) 1 2 d t

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.2, Problem 33E

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To calculate: The general solution of the differential equation, dydt=t23+5t.

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Chapter 8.2, Problem 32E

Chapter 8.2, Problem 34E

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Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing an Antiderivative In Exercises 14, select...Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, select the...Ch. 8.1 - Choosing a Formula In Exercises 5-14, selectthe...

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The general solution of the differential equation, d y d t = t 2 3 + 5 t .

Solution Summary: The author explains the formula used to calculate the general solution of the differential equation, cc

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To calculate: The general solution of the differential equation, dydt=t23+5t.

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