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Find a series expansion, up to xº, for the general solution of Hermite's equation: y" — xy + sy = 0. Write the expansions for the specific solutions y₁, 32, satisfying y₁ (0) = 1, y₁ (0) = 0 and y2 (0) = 0, ; y/₂(0) = 1. Use the recurrence relation to show that the series for either y₁ or for y2 truncates to a polynomial if s is a positive integer. Write down the explicit forms of the expansions for y₁ and y2 for each of the cases s = 3 and s = 4.

Find a series expansion, up to xº, for the general solution of Hermite's equation: y" — xy + sy = 0. Write the expansions for the specific solutions y₁, 32, satisfying y₁ (0) = 1, y₁ (0) = 0 and y2 (0) = 0, ; y/₂(0) = 1. Use the recurrence relation to show that the series for either y₁ or for y2 truncates to a polynomial if s is a positive integer. Write down the explicit forms of the expansions for y₁ and y2 for each of the cases s = 3 and s = 4.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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A3 Find a series expansion, up to x6, for the general solution of Hermite's equation: y" - xy + sy = 0. Write the expansions for the specific solutions y₁, y2, satisfying y₁ (0) = 1,; y₁ (0) = 0 and y2 (0) = 0, ; 3₂(0) = 1. Use the recurrence relation to show that the series for either y₁ or for y2 truncates to a polynomial if s is a positive integer. Write down the explicit forms of the expansions for y₁ and y2 for each of the cases s 3 and s = 4.
Transcribed Image Text:A3 Find a series expansion, up to x6, for the general solution of Hermite's equation: y" - xy + sy = 0. Write the expansions for the specific solutions y₁, y2, satisfying y₁ (0) = 1,; y₁ (0) = 0 and y2 (0) = 0, ; 3₂(0) = 1. Use the recurrence relation to show that the series for either y₁ or for y2 truncates to a polynomial if s is a positive integer. Write down the explicit forms of the expansions for y₁ and y2 for each of the cases s 3 and s = 4.
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