Explanation Given: The differential equation is t 3 d y d t + 3 t 2 y = cos t . Initial condition is y ( π ) = 0 . Integrating factor: I ( t ) = e ∫ P ( t ) d t (1) Product rule: d ( f ⋅ g ) d t = f ⋅ d ( g ) d t + g ⋅ d ( f ) d t Calculation: Obtain the solution of the differential equation t 3 d y d t + 3 t 2 y = cos t . Consider the differential equation t 3 d y d t + 3 t 2 y = cos t . Divide the differential equation by t 3 , t 3 d y d t + 3 t 2 y t 3 = cos t t 3 t 3 d y d t t 3 + 3 t 2 y t 3 = cos t t 3 d y d t + 3 t y = cos t t 3 Thus, the redefined differential equation is d y d t + 3 t y = cos t t 3 . This is a first order linear differential equation of the form y ′ + P ( t ) y = Q ( t ) . Comparing the coefficients of y and constant terms, P ( t ) = 3 t and Q ( t ) = cos t t 3 The integrating factor is computed as follows, Substitute P ( t ) = 3 t in equation (1), I ( t ) = e 3 ∫ 1 t d t = e 3 ln t <

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 9.5, Problem 16E

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To find: The solution of initial value problem.

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Solution Summary: The author explains the solution of the differential equation: y=mathrmsint

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