Explanation Given: The differential equation is x 3 y ″ + y = 0 and x 2 y ″ + ( 3 x − 1 ) y ′ + y = 0 Concept: Frobenius method: Assume y is a function of initial value problem then the solution of the given function can be written in the form y = ∑ n = 0 ∞ C n x n + r . Calculation: The differential equation is given as follows: x 3 y ″ + y = 0 ...... ( 1 ) From the Frobenius method, the given equation can be represented as follows: y = ∑ n = 0 ∞ C n x n + r Now differentiate the above equation. y ′ = ∑ n = 0 ∞ ( n + r ) C n x n + r − 1 Further differentiate the above equation. y ″ = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) C n x n + r − 2 Substitute the value of y and y ″ in equation (1). x 3 y ″ + y = 0 x 3 ( ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) C n x n + r − 2 ) + ( ∑ n = 0 ∞ C n x n + r ) = 0 ( ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) C n x n + r + 1 ) + ( ∑ n = 0 ∞ C n x n + r ) = 0 Now further substitute n = k − 1 in first term and n = k in remaining term, ( ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) C n x n + r + 1 ) + ( ∑ n = 0 ∞ C n x n + r ) = 0 ( ∑ k = 1 ∞ ( k − 1 + r ) ( k − 1 + r − 1 ) C k − 1 x k − 1 + r + 1 ) + c 0 x r + ( ∑ k = 1 ∞ C k x k + r ) = 0 ( ∑ k = 1 ∞ ( k − 1 + r ) ( k + r − 2 ) C k − 1 x k + r ) + c 0 x r + ( ∑ k = 1 ∞ C k x k + r ) = 0 Compare the coefficients of power of x . c 0 = 0 and ( k + r − 1 ) ( k + r − 2 ) c k − 1 + c k = 0 Further simplify for c k such that c k = − ( k + r − 1 ) ( k + r − 2 ) c k − 1 Since c 0 = 0 , other coefficients are also 0 . Thus, the solution is y ( x ) = 0 _ . Similarly the other differential equation is labeled as (2). x 2 y ″ + ( 3 x − 1 ) y ′ + y = 0 ...... ( 2 ) From the Frobenius method, the given equation can be represented as follows: y = ∑ n = 0 ∞ C n x n + r Now differentiate the above equation. y ′ = ∑ n = 0 ∞ ( n + r ) C n x n + r − 1 Further differentiate the above equation. y ″ = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) C n x n + r − 2 Substitute the above value in equation (1)

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 6.3, Problem 36E

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Whether the method of Frobenius yields a series solution of differential equations x3y″+y=0 and x2y″+(3x−1)y′+y=0 at x=0.

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Chapter 6.3, Problem 35E

Chapter 6.3, Problem 37E

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Whether the method of Frobenius yields a series solution of differential equations x 3 y ″ + y = 0 and x 2 y ″ + ( 3 x − 1 ) y ′ + y = 0 at x = 0 .

Solution Summary: The author explains how the Frobenius method yields a series solution of differential equations.

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Whether the method of Frobenius yields a series solution of differential equations x3y″+y=0 and x2y″+(3x−1)y′+y=0 at x=0.

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