Explanation Formula used: The basic form of the differential equation in terms of cosine is, a y ″ + b y ′ + c y = C t m e α t sin ( β t ) …… ( 2 ) Here, the non-negative integer is m and β ≠ 0 . The particular of the above differential equation is given by, y p ( t ) = [ t s ( A m t m + ................ + A 1 t + A 0 ) e α t cos β t + t s ( B m t m + ................ + B 1 t + B 0 ) e α t sin β t ] …… ( 3 ) Here, s will be zero if α + i β is not a root of the associated auxiliary equation, s will be 1 if α + i β is a root of associated auxiliary equation. Calculation: The differential equation is given by, y ″ + 2 y ′ + 2 y = 8 t 3 e − t sin t …… ( 1 ) The auxiliary equation for the equation ( 1 ) . r 2 + 2 r + 2 = 0 Factorize the above equation. r 2 + 2 r + 2 = 0 r 2 + 2 r + 1 + 1 = 0 ( r + 1 ) 2 + 1 = 0 ( r + 1 ) = ± i The roots are, r 1 = − 1 + i r 2 = − 1 − i Hence, two distinct roots are obtained for the auxiliary equation so the value of s is 1 . Compare the differential equation with the equation ( 2 ) . m = 3 α = − 1 β = 1 Substitute 1 for s , 3 for m , − 1 for α and 1 for β equation ( 3 )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 4.4, Problem 31E

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The form of a particular solution for the differential equation y″+2y′+2y=8t3e−tsint.

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Chapter 4.4, Problem 30E

Chapter 4.4, Problem 32E

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Fundamentals of Differential Equations and Boundary Value Problems

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The form of a particular solution for the differential equation y ″ + 2 y ′ + 2 y = 8 t 3 e − t sin t .

Solution Summary: The author explains the form of a particular solution for the differential equation.

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To find:

The form of a particular solution for the differential equation y″+2y′+2y=8t3e−tsint.

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Chapter 4 Solutions

The form of a particular solution for the differential equation y ″ + 2 y ′ + 2 y = 8 t 3 e − t sin t .

Solution Summary: The author explains the form of a particular solution for the differential equation.

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To find: The form of a particular solution for the differential equation y″+2y′+2y=8t3e−tsint.

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Chapter 4 Solutions

To find:

The form of a particular solution for the differential equation y″+2y′+2y=8t3e−tsint.