3. (a) Show that x = 0 is a regular singular point for the Laguerre equation xy" + (1 − x)y' + 2y = 0, and using the Frobenius method, show that both roots of the indicial equation are equal to zero. Show that the corresponding series for the solution y = Σ anx" is a polynomial and find its explicit form. n=0 (b) Give an example of a second order linear differential equation with polynomial coefficients possessing exactly four singular points such that the points x = ±2 are regular singular points and the points x = ±1 are irregular singular points. Justify your example. (c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.
3. (a) Show that x = 0 is a regular singular point for the Laguerre equation xy" + (1 − x)y' + 2y = 0, and using the Frobenius method, show that both roots of the indicial equation are equal to zero. Show that the corresponding series for the solution y = Σ anx" is a polynomial and find its explicit form. n=0 (b) Give an example of a second order linear differential equation with polynomial coefficients possessing exactly four singular points such that the points x = ±2 are regular singular points and the points x = ±1 are irregular singular points. Justify your example. (c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.
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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Transcribed Image Text:3. (a) Show that x = 0 is a regular singular point for the Laguerre equation
xy" + (1 − x)y' + 2y = 0,
and using the Frobenius method, show that both roots of the indicial equation are
equal to zero. Show that the corresponding series for the solution y = Σ anx" is
a polynomial and find its explicit form.
n=0
(b) Give an example of a second order linear differential equation with polynomial
coefficients possessing exactly four singular points such that the points x = ±2 are
regular singular points and the points x = ±1 are irregular singular points. Justify
your example.
(c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution
y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.
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