05series_solution2_frobenius_examples

# 05series_solution2_frobenius_examples - Series Solutions to...

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Series Solutions to Linear Ordinary Differential Equations III Examples: Frobenius' Solution about Regular Singular Points 1. Text Example 2, pp. 253, 3rd ed Text Example 2, pp. xxx, 2nd ed ODE: 0 y ' y ' ' xy 3 = - + Standard form: 0 y x 3 1 ' y x 3 1 ' ' y = - + ( 29 x 3 1 x P = ( 29 x 3 1 x Q - = Singular point at x = 0 ( 29 ( 29 3 1 x 3 x x xP x p = = = SP removed ( 29 ( 29 3 x x 3 x x Q x x q 2 2 - = - = = SP removed Regular singular point Use method of Frobenius ( 29 + + + + + = = = + + + + = + = 4 r 4 3 r 3 2 r 2 1 r 1 r 0 0 n r n n 0 n n n r x c x c x c x c x c x c x c x x y Derivatives ( 29 = - + + = 0 n 1 r n n x c r n ' y ( 29 ( 29 ( 29 ( 29 + + + + + + + + + = + + + - 3 r 4 2 r 3 1 r 2 r 1 1 r 0 x c 4 r x c 3 r x c 2 r x c 1 r x rc ( 29 ( 29 = - + - + + = 0 n 2 r n n x c 1 r n r n ' ' y ( 29 ( 29 ( 29 ( 29 r 2 1 r 1 2 r 0 x c 1 r 2 r x c 1 r r x c 1 r r + + + + + - = - - ( 29 ( 29 ( 29 ( 29 + + + + + + + + + 2 r 4 1 r 3 x c 3 r 4 r x c 2 r 3 r Substitute ( 29 ( 29 ( 29 ( 29 r 2 1 r 1 2 r 0 x c 1 r 2 r x 3 x c 1 r xr 3 x c 1 r xr 3 + + + + + - - - ( 29 ( 29 ( 29 ( 29 + + + + + + + + + 2 r 4 1 r 3 x c 3 r 4 r x 3 x c 2 r 3 r x 3 ( 29 ( 29 ( 29 ( 29 + + + + + + + + + + + + + - 3 r 4 2 r 3 1 r 2 r 1 1 r 0 x c 4 r x c 3 r x c 2 r x c 1 r x rc - - - - - - + + + + 4 r 4 3 r 3 2 r 2 1 r 1 r 0 x c x c x c x c x c Multiply through ( 29 ( 29 ( 29 ( 29 1 r 2 r 1 1 r 0 x c 1 r 2 r 3 x c 1 r r 3 x c 1 r r 3 + - + + + + + - ( 29 ( 29 ( 29 ( 29 + + + + + + + + + 3 r 4 2 r 3 x c 3 r 4 r 3 x c 2 r 3 r 3 ( 29 ( 29 ( 29 ( 29 + + + + + + + + + + + + + - 3 r 4 2 r 3 1 r 2 r 1 1 r 0 x c 4 r x c 3 r x c 2 r x c 1 r x rc - - - - - - + + + + 4 r 4 3 r 3 2 r 2 1 r 1 r 0 x c x c x c x c x c

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2 Series Solutions to Linear Ordinary Differential Equations III Examples: Frobenius' Solution about Regular Singular Points Collect on like powers of x ( 29 [ ] ( 29 ( 29 [ ] r 0 1 1 1 r 0 0 x c c 1 r c 1 r r 3 x rc c 1 r r 3 - + + + + + - - ( 29 ( 29 ( 29 [ ] 1 r 1 2 2 x c c 2 r c 1 r 2 r 3 + - + + + + + ( 29 ( 29 ( 29 [ ] 2 r 2 3 3 x c c 3 r c 2 r 3 r 3 + - + + + + + ( 29 ( 29 ( 29 [ ] + - + + + + + + 3 r 3 4 4 x c c 4 r c 3 r 4 r 3 Set coefficients equzl to zero term by term 1 r x - : ( 29 ( 29 ( 29 0 2 r 3 r c 1 3 r 3 r c 0 rc c 1 r 3 0 0 0 0 = - = + - = + - Indicial equation
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05series_solution2_frobenius_examples - Series Solutions to...

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