Explanation Given: The given differential equation is, x 2 y ' ' − x ( 1 − x ) y ' + ( 1 − x ) y = 0 Approach: Consider the differential equation, x 2 y ' ' + x p ( x ) y ' + q ( x ) y = 0 where p and q are analytic at x = 0 . Suppose that, p ( x ) = ∑ n = 0 ∞ p n x n , q ( x ) = ∑ n = 0 ∞ q n x n . for | x | < R . Let r 1 and r 2 denote the roots of the indicial equation and assume that r 1 ≥ r 2 if these roots are real. Then has two linearly independent solutions valid on the interval ( 0 , R ) . The form of solution is determined as follows: 1. r 1 − r 2 not an integer: y 1 ( x ) = x r 1 ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = x r 2 ∑ n = 0 ∞ b n x n , b 0 ≠ 0 2. r 1 = r 2 = r : y 1 ( x ) = x r ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = y 1 ( x ) ln x + x r ∑ n = 1 ∞ b n x n 3. r 1 − r 2 equals a positive integer: y 1 ( x ) = x r 1 ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = A y 1 ( x ) ln x + x r 2 ∑ n = 0 ∞ b n x n , b 0 ≠ 0 The coefficient in each of these solutions can be determined by direct substitution in the considered differential equation. Finally, if x r 1 and x r 2 are replaced by | x | r 1 and | x | r 2 , respectively, we obtain the linearly independent solutions that are valid for ( 0 , R ) . Calculation: The standard form of differential equation is given as, x 2 y ' ' + x p ( x ) y ' + q ( x ) y = 0 … ( 1 ) The indicial equation is given as, r ( r − 1 ) + p ( 0 ) r + q ( 0 ) = 0 … ( 2 ) where r is a root of the indicial equation. Consider the given differential equation, x 2 y ' ' − x ( 1 − x ) y ' + ( 1 − x ) y = 0 … ( 3 ) Compare equation ( 1 ) and ( 3 ) . p ( x ) = − ( 1 − x ) And, q ( x ) = ( 1 − x ) Thus, p ( 0 ) = − ( 1 − 0 ) = − 1 And, q ( 0 ) = ( 1 − 0 ) = 1 Substitute − 1 for p ( 0 ) and 1 for q ( 0 ) in equation ( 2 ) . r ( r − 1 ) + ( − 1 ) r + ( 1 ) = 0 r 2 − r − r + 1 = 0 r 2 − 2 r + 1 = 0 r ( r − 1 ) − 1 ( r − 1 ) = 0 Further solve the above equation. ( r − 1 ) ( r − 1 ) = 0 r = 1 , 1 Since 1 is a repeated root, the general form of two linearly independent solutions are, y 1 ( x ) = x r ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = y 1 ( x ) ln x + x r ∑ n = 1 ∞ b n x n Substitute 1 for r in the above expressions to obtain the general form of two linearly independent solutions to the differential equation on an interval ( 0 , R ) . y 1 ( x ) = x 1 ∑ n = 0 ∞ a n x n And, y 2 ( x ) = y 1 ( x ) ln x + x 1 ∑ n = 1 ∞ b n x n . Determine the coefficients. y ( x ) = x r ∑ n = 0 ∞ a n x n = ∑ n = 0 ∞ a n x n + r Differentiate above equation twice with respect to x . y ′ ( x ) = ∑ n = 0 ∞ ( r + n ) a n x r + n − 1 y ″ ( x ) = ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n x r + n − 2 Substitute the value of y ″ , y ′ and y in the given equation. [ x 2 ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n x r + n − 2 − x ( 1 − x ) ∑ n = 0 ∞ ( r + n ) a n x r + n − 1 + ( 1 − x ) ∑ n = 0 ∞ a n x n + r = 0 ] [ ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n x r + n − ( 1 − x ) ∑ n = 0 ∞ ( r + n ) a n x r + n + ( 1 − x ) ∑ n = 0 ∞ a n x r + n = 0 ] ∑ n = 0 ∞ [ ( r + n ) ( r + n − 1 ) − ( r + n ) + 1 ] a n x n + r + ∑ n = 0 ∞ ( r + n − 1 ) a n x r + n + 1 = 0 Dividing by x r and replace n by n − 1 in the second term. ∑ n = 0 ∞ [ ( r + n ) ( r + n − 2 ) + 1 ] a n x n + ∑ n = 0 ∞ ( r + n − 2 ) a n − 1 x n = 0 The remaining coefficient must satisfy the recurrence relation. [ ( r + n ) ( r + n − 2 ) + 1 ] a n + ( r + n − 2 ) a n − 1 = 0 , n = 1 , 2 , 3 , ..... a n = − ( r + n − 2 ) [ ( r + n ) ( r + n − 2 ) + 1 ] a n − 1 For r = 1 in the above equation, a n = − ( n − 1 ) n 2 a n − 1 , n = 1 , 2 , 3 ,

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ISBN: 9780321990167

Author: ANNIN

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 11.5, Problem 25P

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The two linearly independent solutions to the given differential equation x2y''+xp(x)y'+q(x)y=0 on (0,∞).

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EBK DIFFERENTIAL EQUATIONS AND LINEAR A

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The two linearly independent solutions to the given differential equation x 2 y ' ' + x p ( x ) y ' + q ( x ) y = 0 on ( 0 , ∞ ) .

Solution Summary: The author explains that the two linearly independent solutions to the given differential equation are x2y''+xp

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The two linearly independent solutions to the given differential equation x2y''+xp(x)y'+q(x)y=0 on (0,∞).

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