Explanation Procedure used: General solution of a differential equation by undetermined coefficients: 1. Consider the given differential equation a 1 y ( n ) + ... + a n − 1 y ′ + a n y = g ( x ) and determine the homogeneous differential equation associated with the DE. 2. Calculate the roots of auxiliary equation of homogeneous equation. 3. Determine g ( x ) . 4. Assume the particular solution to the differential equation in form of an equation. 5. Substitute, the particular solution into the differential equation. 6. Compare the coefficients and determine the general solution. 7. Apply the initial values and determine the unknown coefficients. 8. Substitute, the values of coefficients of complimentary solution into the general solution to find the solution of differential equation with initial values. Calculation: Consider differential equation y ″ + y = x 2 + 1 . The associate homogenous equation of differential equation y ″ + y = 0 . The auxiliary equation corresponding to the homogenous equation, y ″ + y = 0 is m 2 + 1 = 0 . Find the roots of m 2 + 1 = 0 . Therefore, m 2 + 1 = 0 m 2 = − 1 m = ± i Hence the complimentary function is y c = ( c 1 cos x + c 2 sin x ) Compare the differential equation y ″ + y = x 2 + 1 to the y ″ + y = g ( x ) . Therefore, g ( x ) = x 2 + 1 . Assume that a particular solution of the given differential equation is also in the form of a quadratic function y p = A x 2 + B x + C . y p = A x 2 + B x + C y ′ p = 2 A x + B y ″ p = 2 A Substitute these values into the differential equation. Therefore, y ″ p + y p = x 2 + 1 2 A + A x 2 + B x + C = x 2 + 1 Equate the coefficient of x 2 . Thus, A = 1 Equate the coefficient of x . Thus, B = 0 Equate the coefficient of constant. Thus, 2 A + C = 1 2 ⋅ 1 + C = 1 2 + C = 1 C = − 1 Substitute all the values into the particular solution. Therefore, y p = A x 2 + B x + C y p = x 2 + 0

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.4, Problem 36E

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

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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