power_series_ode_solution1

Power_series_ode_sol - Series Solutions to Linear Ordinary Differential Equations I Power Series Solution for the Harmonic Oscillator Equation ODE

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Power Series Solution for the Harmonic Oscillator Equation : ODE: 0 y dx y d 2 2 2 = λ + Transform independent variable x λ = ς Transform derivatives ς λ = ς ς = d dy dx d d dy dx dy 2 2 2 2 2 d y d dx d d dy d d d dy dx d dx dy dx d dx y d ς λ = ς ς λ ς = ς λ = = Substitute in ODE: 0 y d y d y dx y d 2 2 2 2 2 2 2 = λ + ς λ = λ + Simplify 0 y d y d 2 2 = + ς Express dependent variable as a power series Function ( 29 + + + + + + + = = = 7 7 6 6 5 5 4 4 3 3 2 2 1 0 0 n n n ς c ς c ς c ς c ς c ς c ς c c ς c ς y Derivatives: ( 29 + + + + + + = = = - 5 6 4 5 3 4 2 3 2 1 1 n 1 n n ς c 6 ς c 5 ς c 4 ς c 3 ς c 2 ς c ς nc ς d ς dy ( 29 ( 29 = - - = ⋅ ⋅ + + + + + = 2 n 2 n n 5 7 4 6 3 5 2 4 3 2 2 2 ς 1 n n c ς c 6 7 ς c 5 6 ς c 4 5 ς c 3 4 ς c 2 3 c 1 2 ς d ς y d ( 29 ( 29 0 ς c ς c ς c ς c ς c ς c ς c c ς c 6 7 ς c 5 6 ς c 4 5 ς c 3 4 ς c 2 3 c 1 2 y ς d ς y d 7 7 6 6 5 5 4 4 3 3 2 2 1 0 5 7 4 6 3 5 2 4 3 2 2 2 = + + + + + + + + + + + + + = + Substitute in ODE and collect terms on powers of ς ( 29 ( 29 ( 29 ( 29 ( 29 0 ς c c 6 7 ς c c 5 6 ς c c 4 5 ς c c 3 4 ς c c 2 3 c c 1 2 5 5 7 4 4 6 3 3 5 2 2 4 1 3 0 2 = + + + + + + + + + + + + The coefficients of each power of ς must equal 0 to satisfy the equation for all ς Equating these terms 0 ς : 1 2 c c 0 c c 1 2 0 2 0 2 - = = + 1 ς : 2 3 c c 0 c c 2 3 1 3 1 3 - = = + 2 ς : 1 2 3 4 c 3 4 c c 0 c c 3 4 0 2 4 2 4 = - = = + 3 ς : 1 2 3 4 5 c 4 5 c c 0 c c 4 5 1 3 5 3 5 = -

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This note was uploaded on 04/11/2008 for the course CHNE 525 taught by Professor Prinja during the Fall '08 term at New Mexico.

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Power_series_ode_sol - Series Solutions to Linear Ordinary Differential Equations I Power Series Solution for the Harmonic Oscillator Equation ODE

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