Explanation Given: The differential equation is ( x − 1 ) y ″ + 3 y = 0 . Calculation: The differential equation is given as follows: ( x − 1 ) y ″ + 3 y = 0 ...... ( 1 ) Suppose that y = ∑ n = 0 ∞ c n x n . Since x = 0 is regular singular point, differentiate the above equation. y ' = ∑ n = 1 ∞ n c n x n − 1 Again differentiate the above equation. y ″ = ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 Substitute the value of y ′ and y ″ in the equation (1). ( x − 1 ) y ″ + 3 y = 0 ( x − 1 ) ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 + 3 ∑ n = 0 ∞ c n x n + = 0 ∑ n = 2 ∞ n ( n − 1 ) c n ︸ k = n − 1 x n − 1 − ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 ︸ k = n − 2 + 3 ∑ n = 0 ∞ c n x n ︸ k = n = 0 Further simplify the above equation. ∑ k = 1 ∞ k ( k + 1 ) c k + 1 x k − ∑ k = 0 ∞ ( k + 2 ) ( k + 1 ) c k + 2 x k + 3 ∑ k = 0 ∞ c k x k = 0 ∑ k = 1 ∞ k ( k + 1 ) c k + 1 x k − 2 c 2 − ∑ k = 1 ∞ ( k + 2 ) ( k + 1 ) c k + 2 x k + 3 c 0 + 3 ∑ k = 1 ∞ c k x k = 0 3 c 0 − 2 c 2 + ∑ k = 1 ∞ [ k ( k + 1 ) c k + 1 − ( k + 2 ) ( k + 1 ) c k + 2 + 3 c k ] x k = 0 Compare the coefficients, the first relation is obtained as follows: 3 c 0 − 2 c 2 = 0 ...... ( 2 ) After the comparison the second relation is obtained as follows: k ( k + 1 ) c k + 1 − ( k + 2 ) ( k + 1 ) c k + 2 + 3 c k = 0 c k + 2 = k ( k + 2 ) c k + 1 + 3 ( k + 2 ) ( k + 1 ) c k

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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Chapter 6, Problem 11RE

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Two solutions of the differential equation (x−1)y″+3y=0 with the use of appropriate infinite series method about x=0.

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Exercise 11.3 A slope field is given for the equation y' = 4y+4. (a) Sketch the particular solution that corresponds to y(0) = −2 (b) Find the constant solution (c) For what initial conditions y(0) is the solution increasing? (d) For what initial conditions y(0) is the solution decreasing? (e) Verify these results using only the differential equation y' = 4y+4.

SOLVE ONLY FOR (L) (M) AND (O)

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Two solutions of the differential equation ( x − 1 ) y ″ + 3 y = 0 with the use of appropriate infinite series method about x = 0 .

Solution Summary: The author calculates the differential equation (x-1)y′′+3y=0 with the use of appropriate infinite series method.

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Two solutions of the differential equation (x−1)y″+3y=0 with the use of appropriate infinite series method about x=0.

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