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# 3_2 - Solutions 3.2-Page 215 Problem 1 Find the general...

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Solutions 3.2-Page 215 Problem 1 Find the general solutions in powers of x of the differential equations. State the recurrence relations and the guaranteed radius of convergence in each case. 0 2 4 ) 1 ( 2 = + + y y x y x Substituting , , and into the differential equation yields = = 0 n n n x c y = = 1 1 n n n x nc y = = 2 2 ) 1 ( n n n x c n n y 0 2 4 ) 1 ( ) 1 ( 0 1 1 2 2 2 = + + = = = n n n n n n n n n x c x nc x x c n n x Simplifying further, 0 2 4 ) 1 ( ) 1 ( 0 1 2 2 2 = + + = = = = n n n n n n n n n n n n x c x nc x c n n x c n n The first and third summations can start at n = 0 and no additional nonzero terms will be added. However, the second summation must be rewritten to start at n = 0 such that the identity principle can be used. 0 2 4 ) 1 )( 2 ( ) 1 ( 0 0 0 2 0 = + + + + = = = + = n n n n n n n n n n n n x c x nc x c n n x c n n Using the identity principle and summing coefficients yields [ ] [ ] ) 1 )( 2 ( ) 1 )( 2 ( ) 1 )( 2 ( 2 3 ) 1 )( 2 ( 2 4 ) 1 ( ) 1 )( 2 ( 2 4 ) 1 ( 0 2 4 ) 1 )( 2 ( ) 1 ( 2 2 2 + + + + = + + + + = + + + + = + + + + = = + + + + + + n n n n c n n n n c n n n n n c n n c nc c n n c c nc c n n c n n n n n n n n n n n n n So the recurrence formula is n n c c = + 2 From this recurrence formula, it is evident that coefficients with even indices (n = 0,2,4…) are equal to , and coefficients with odd indices (n = 1,3,5…) are equal to . 0 c 1 c

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Therefore = + = = + = = 0 1 2 1 0 2 0 0 n n n n n n n x c x c x c y Expanding the summations gives [ ] [ [ ] [ ] ... 1 ... 1 ... ... 1 4 2 1 4 2 0 5 3 1 4 2 0 0 1 2 1 0 2 0 x x x c x x c y x x x c x x c y x c x c y n n n n + + + + + = + + + + + = + = = + = ] The first summations match the form of x 1 1 if the variable is instead of 2 x x . So 2 1 2 0 1 1 x x c x c y + = 2 1 0 1 x x c c y + = The notation of the text is to write a differential equation as . For this problem, , which has a singular point at x = 1, -1. The distance from one of these points to a = 0 is 1, so the guaranteed radius of convergence is at least 1. 0 ) ( ) ( ) ( = + + y x C y x B y x A 1 ) ( 2 = x x A 1 ρ
Problem 3 Find the general solutions in powers of x of the differential equations. State the recurrence relations and the guaranteed radius of convergence in each case. 0

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3_2 - Solutions 3.2-Page 215 Problem 1 Find the general...

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