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3. Sometimes one can change an irregular singular point to a regular singular point, by a suitable change of variable, so that the Frobenius method can be applied. a) Show that x = 0 is indeed an irregular singular point for y "+√xy = 0 (x > 0). b) Show that if we change the independent variable from "x" to "t" according to √x =t, then the original equation on y (x(t)) =Y (t) converts to: Y "(t)--Y '(t)+4t³Y (t) = 0 = (t > 0) c) Show that the converted ODE has a regular singular point at t = 0 (t = 0 is corresponding value at x = 0). d) Find a general solution for converted ODE by the Frobenius method. Give the general term of series coefficients if possible. e) Plug in t√x in the result to obtain the corresponding general solution of y "+ √x y = 0 = (x >0).

3. Sometimes one can change an irregular singular point to a regular singular point, by a suitable change of variable, so that the Frobenius method can be applied. a) Show that x = 0 is indeed an irregular singular point for y "+√xy = 0 (x > 0). b) Show that if we change the independent variable from "x" to "t" according to √x =t, then the original equation on y (x(t)) =Y (t) converts to: Y "(t)--Y '(t)+4t³Y (t) = 0 = (t > 0) c) Show that the converted ODE has a regular singular point at t = 0 (t = 0 is corresponding value at x = 0). d) Find a general solution for converted ODE by the Frobenius method. Give the general term of series coefficients if possible. e) Plug in t√x in the result to obtain the corresponding general solution of y "+ √x y = 0 = (x >0).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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3. Sometimes one can change an irregular singular point to a regular singular point, by a suitable change of variable, so that the Frobenius method can be applied. a) Show that x = 0 is indeed an irregular singular point for y "+√xy = 0 (x > 0). b) Show that if we change the independent variable from "x" to "t" according to √x =t, then the original equation on y (x(t)) =Y (t) converts to: Y "(t)--Y '(t)+4t³Y (t) = 0 = (t > 0) c) Show that the converted ODE has a regular singular point at t = 0 (t = 0 is corresponding value at x = 0). d) Find a general solution for converted ODE by the Frobenius method. Give the general term of series coefficients if possible. e) Plug in t√x in the result to obtain the corresponding general solution of y "+ √x y = 0 = (x >0).
Transcribed Image Text:3. Sometimes one can change an irregular singular point to a regular singular point, by a suitable change of variable, so that the Frobenius method can be applied. a) Show that x = 0 is indeed an irregular singular point for y "+√xy = 0 (x > 0). b) Show that if we change the independent variable from "x" to "t" according to √x =t, then the original equation on y (x(t)) =Y (t) converts to: Y "(t)--Y '(t)+4t³Y (t) = 0 = (t > 0) c) Show that the converted ODE has a regular singular point at t = 0 (t = 0 is corresponding value at x = 0). d) Find a general solution for converted ODE by the Frobenius method. Give the general term of series coefficients if possible. e) Plug in t√x in the result to obtain the corresponding general solution of y "+ √x y = 0 = (x >0).
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