Explanation Procedure used: The general solution of the homogeneous differential equation a y ″ + b y ′ + c y = G ( x ) can be written as y ( x ) = y c ( x ) + y p ( x ) , where y c ( x ) is complimentary solution of a y ″ + b y ′ + c y = 0 and y p ( x ) is the particular solution of a y ″ + b y ′ + c y = G ( x ) The complimentary solution is y c = C 1 y 1 ( x ) + C 2 y 2 ( x ) . (1) Here C 1 and C 2 are constants. The parameters C 1 and C 2 in y c can be replaced with functions u 1 and u 2 (variable parameters), so that y p ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) (2) Now determine two unknown functions u 1 and u 2 , by Cramer’s Rule, y 1 u ′ 1 + y 2 u ′ 2 = 0 (3) y 1 u ′ 1 + y 2 u ′ 2 = f ( x ) (4) Using Wronskian method, W = | y 1 y 2 y ′ 1 y ′ 2 | (5) W 1 = | 0 y 2 f ( x ) y ′ 2 | , W 2 = | y 1 0 y ′ 1 f ( x ) | ( u ′ 1 = W 1 W = − y 2 f ( x ) W ) and ( u ′ 2 = W 2 W = y 1 f ( x ) W ) (6) The functions u 1 and u 2 can be found by integrating the results in (6). Substitute the values of u 1 and u 2 in equation (2) to find the particular solution. The general solution of the equation is then y = y c + y p . Given: The given differential equation is y ″ + 3 y ′ + 2 y = 1 1 + e x . Calculations: The auxiliary equation for the associated homogeneous equation is m 2 + 3 m + 2 = 0 . m 2 + 3 m + 2 = 0 ( m + 1 ) ( m + 2 ) = 0 m 1 = − 1 and m 2 = − 2 The complementary solution is y c ( x ) = C 1 e − x + C 2 e − 2 x

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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The general solution of the given differential equation by variation of parameters.

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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The general solution of the given differential equation by variation of parameters.

Solution Summary: The author explains the general solution of the homogeneous differential equation ay′′+3xprime by variation of parameters.

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