Explanation Given: The given initial boundary value problem is: x 2 y ″ + x y ′ + y = 4 x − 1 ; y ( 1 ) = 2 , y ( e π ) = 2 e − π Procedure used: Solution for nonhomogeneous equation: Consider the equation, a y ″ + b y ′ + c y = q x . First find the solution of the auxiliary equation: a r 2 + b r + c = 0 If the roots are conjugate complex α ± β i , then the general solution form is: y ( x ) = c 1 e α x cos β x + c 2 e α x sin β x After that, use the initial condition to find the value of the constants in the general solution. Undetermined coefficients Method for finding the form of particular solution: Consider the non-homogeneous system X ′ ( t ) = A X ( t ) + f ( t ) Here, A is an n × n constant matrix and the entries of f ( t ) are polynomials, exponential, sine’s etc. Choosing the form of particular solution: If the column of f ( t ) contain x − 1 , the particular solution form is a x − 1 . Here, a is the constants. Calculation: Consider the given initial boundary value problem: x 2 y ″ + x y ′ + y = 4 x − 1 ; y ( 1 ) = 2 , y ( e π ) = 2 e − π Substitute y = x r . r 2 + 1 = 0 r = ± i The general solution is: y ( x ) = c 1 cos ( ln x ) + c 2 sin ( ln x ) Find the particular form by using the undetermined coefficients Method: The particular solution form for x − 1 : y p ( x ) = a x − 1 Substituting the y p ( x ) in the original equation x 2 y ″ + x y ′ + y = 4 x − 1 , we get:

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 11.2, Problem 11E

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The solution for the given initial boundary value problem x2y″+xy′+y=4x−1; y(1)=2, y(eπ)=2e−π.

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Chapter 11.2, Problem 10E

Chapter 11.2, Problem 12E

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Fundamentals of Differential Equations and Boundary Value Problems

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

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The solution for the given initial boundary value problem x 2 y ″ + x y ′ + y = 4 x − 1 ; y ( 1 ) = 2 , y ( e π ) = 2 e − π .

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The solution for the given initial boundary value problem x2y″+xy′+y=4x−1; y(1)=2, y(eπ)=2e−π.

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Chapter 11 Solutions