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The Hermite equation is a second order linear homogeneous ordinary differential equation, y" - 2xy + y = 0, where A is a real number. We are interested in series solutions of the form y(x) = 0 an xn (a) Show that the recurrence relation for the expansion coefficients an is given by n=0 an+2= 2n - A (n+1)(n+2) ·an· (b) Explain why the series solution has two free parameters. Show that series solutions Yeven with ao 0, a₁ = 0 are even functions Show that series solutions yodd with ao = 0, a₁ #0 are odd functions. (c) Set λ = 6. Compute yodd and explain why the series terminates. Explain why the series Yeven does not terminate.

The Hermite equation is a second order linear homogeneous ordinary differential equation, y" - 2xy + y = 0, where A is a real number. We are interested in series solutions of the form y(x) = 0 an xn (a) Show that the recurrence relation for the expansion coefficients an is given by n=0 an+2= 2n - A (n+1)(n+2) ·an· (b) Explain why the series solution has two free parameters. Show that series solutions Yeven with ao 0, a₁ = 0 are even functions Show that series solutions yodd with ao = 0, a₁ #0 are odd functions. (c) Set λ = 6. Compute yodd and explain why the series terminates. Explain why the series Yeven does not terminate.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The Hermite equation is a second order linear homogeneous ordinary differential equation, y" − 2x y' + λy = 0, where is a real number. We are interested in series solutions of the form y(x) =n=0 an xn (a) Show that the recurrence relation for the expansion coefficients an is given by an+2 = 2n - X (n + 1)(n+2) ·an· (b) Explain why the series solution has two free parameters. Show that series solutions Yeven with ao 0, a₁ 0 are even functions Show that series solutions yodd with ao = 0, a₁ #0 are odd functions. (c) Set λ = 6. Compute yodd and explain why the series terminates. Explain why the series Yeven does not terminate.
Transcribed Image Text:The Hermite equation is a second order linear homogeneous ordinary differential equation, y" − 2x y' + λy = 0, where is a real number. We are interested in series solutions of the form y(x) =n=0 an xn (a) Show that the recurrence relation for the expansion coefficients an is given by an+2 = 2n - X (n + 1)(n+2) ·an· (b) Explain why the series solution has two free parameters. Show that series solutions Yeven with ao 0, a₁ 0 are even functions Show that series solutions yodd with ao = 0, a₁ #0 are odd functions. (c) Set λ = 6. Compute yodd and explain why the series terminates. Explain why the series Yeven does not terminate.
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