frobenius_p3

frobenius_p3 - n 1 2 2 x x b x ln x Cy x y ; b Note: For...

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Series Solutions to Linear Ordinary Differential Equations III 3 Method of Frobenius Note: The indicial equation yields tow values for r. These are labeled 1 r and 2 r . By convention we take 1 r to be the larger root 2 1 r r . We get some information about the solutions by looking at the difference between the roots of the indicial equation 2 1 r r - Case 1: Two series solutions 1 r and 2 r are distinct and do not differ by an integer (including 0) Two series solutions exist: ( 29 ( 29 = + - = 0 n r n 0 n 1 1 x x c x y and ( 29 ( 29 = + - = 0 n r n 0 n 2 2 x x b x y In terms of the difference: [ I r r 2 1 - Where = , 3 , 2 , 1 , 0 I ] For 0 x 0 = : ( 29 = + = 0 n r n n 1 1 x c x y and ( 29 = + = 0 n r n n 2 2 x b x y Case 2: Roots differ by a positive integer: N r r 2 1 = - Where = , 3 , 2 , 1 N Two solutions exist: ( 29 ( 29 = + - = 0 n r n 0 n 1 1 x x c x y ; 0 c 0 and ( 29 ( 29 ( 29 = + - + = 0 n r n
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Unformatted text preview: n 1 2 2 x x b x ln x Cy x y ; b Note: For this case we have a possibility that the second solution may have C = This would give 2 series solutions, but we have no way of knowing this before we have solved the ODE completely Case 3: Roots are equal 2 1 2 1 r r r r = =-Two solutions exist: ( 29 ( 29 = +-= n r n n 1 1 x x c x y ; c and ( 29 ( 29 ( 29 = +-+ = n r n n 1 2 2 x x b x ln x y x y Note: The log term always occurs in this case. Finding the second solution: We can use Reduction of Order to find the second solution....
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