Explanation Given: The differential equation is x ( x − 1 ) y ″ + 3 y ' − 2 y = 0 . Calculation: The differential equation is given as follows: x ( x − 1 ) y ″ + 3 y ' − 2 y = 0 ...... ( 1 ) Suppose that y = ∑ n = 0 ∞ c n x n + r . Differentiate the above equation. y ' = ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 Again 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 ( x − 1 ) y ″ + 3 y ' − 2 y = 0 x ( x − 1 ) ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 + 3 ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 − 2 ∑ n = 0 ∞ c n x n + r = 0 [ ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 1 + 3 ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 − 2 ∑ n = 0 ∞ c n x n + r ] = 0 Further simplify the above equation. [ ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 1 + 3 ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 − 2 ∑ n = 0 ∞ c n x n + r ] = 0 ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r ︸ n = k − 1 − ∑ n = 0 ∞ ( n + r ) ( n + r − 4 ) c n x n + r − 1 ︸ n = k − 2 ∑ n = 0 ∞ c n x n + r ︸ n = k − 1 = 0 ∑ k = 1 ∞ ( k + r − 1 ) ( k + r − 2 ) c k − 1 x k + r − 1 − ∑ k = 0 ∞ ( k + r ) ( k + r − 4 ) c k x k + r − 1 − 2 ∑ k = 1 ∞ c k − 1 x k + r − 1 = 0 − r ( r − 4 ) c 0 x r − 1 + x k + r − 1 { ∑ k = 1 ∞ ( k + r − 1 ) ( k + r − 2 ) c k − 1 − ∑ k = 1 ∞ ( k + r ) ( k + r − 4 ) c k − 2 ∑ k = 1 ∞ c k − 1 } = 0 Compare the coefficients. r ( r − 4 ) = 0 { ( k + r − 1 ) ( k + r − 2 ) − 2 } c k − 1 − ( k + r ) ( k + r − 4 ) c k = 0 Two values of r are obtained as 0 , 4

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

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Chapter 6.3, Problem 32E

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The number of solutions of the differential equation x(x−1)y″+3y'−2y=0 with the use of recurrence relation by the method of Frobenius.

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Chapter 6.3, Problem 31E

Chapter 6.3, Problem 33E

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Chapter 6 Solutions

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

Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 10E

Ch. 6.1 - Prob. 11E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 13E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 16E Ch. 6.1 - In Problems 17 and 18 use an appropriate series in...Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - Prob. 28E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - In Problems 3538 proceed as in Example 4 and find...Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.2 - In Problems 1 and 2 without actually solving the...Ch. 6.2 - Prob. 2E Ch. 6.2 - In Problems 3–6 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - Prob. 9E Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Prob. 13E Ch. 6.2 - Prob. 14E Ch. 6.2 - Prob. 15E Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Prob. 18E Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Prob. 23E Ch. 6.2 - In Problems 23 and 24 use the procedure in Example...Ch. 6.2 - Without actually solving the differential equation...Ch. 6.2 - How can the power series method be used to solve...Ch. 6.2 - Prob. 27E Ch. 6.2 - Prob. 28E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 9E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 11E Ch. 6.3 - Prob. 12E Ch. 6.3 - In Problems 13 and 14, x = 0 is a regular singular...Ch. 6.3 - Prob. 14E Ch. 6.3 - In Problems 1524, x = 0 is a regular singular...Ch. 6.3 - Prob. 16E Ch. 6.3 - Prob. 17E Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Prob. 23E Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - In Problems 1320 use (20) to find the general...Ch. 6.4 - Prob. 21E Ch. 6.4 - Assume that b in equation (20) can be pure...Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Use the recurrence relation in Problem 28 along...Ch. 6.4 - Prob. 33E Ch. 6.4 - Prob. 34E Ch. 6.4 - Use the change of variables s=2kmet/2 to show that...Ch. 6.4 - Prob. 36E Ch. 6.4 - Use the result in parts (a) and (b) of Problem 36...Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - (a) Use the explicit solutions y1(x) and y2(x) of...Ch. 6.4 - Prob. 47E Ch. 6.4 - Prob. 48E Ch. 6.4 - Find the first three positive values of for which...Ch. 6.4 - The differential equation y 2xy + 2y = 0 is known...Ch. 6.4 - (a) When = n is a nonnegative integer, Hermites...Ch. 6.4 - Prob. 55E Ch. 6.4 - Prob. 56E Ch. 6 - In Problems 1 and 2 answer true or false without...Ch. 6 - Prob. 2RE Ch. 6 - Prob. 3RE Ch. 6 - Prob. 4RE Ch. 6 - Prob. 5RE Ch. 6 - Prob. 6RE Ch. 6 - Prob. 7RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 9RE Ch. 6 - Prob. 10RE Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 16RE Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 20RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Prob. 24RE Ch. 6 - Prob. 25RE Ch. 6 - Express the general solution of the given...Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE

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The number of solutions of the differential equation x ( x − 1 ) y ″ + 3 y ' − 2 y = 0 with the use of recurrence relation by the method of Frobenius.

Solution Summary: The author compares the number of solutions of the differential equation x(x-1)y′′+3y

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The number of solutions of the differential equation x(x−1)y″+3y'−2y=0 with the use of recurrence relation by the method of Frobenius.

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Chapter 6 Solutions