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Consider the second order linear differential equation z²y″(z) + zy′(z) + (z² − n²)y(z) = 0 - (4) for any fixed n = {0, 1, 2, ...}. a Is zo = 0 an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. b. Consider a power series solution around zo = 0 of the form ∞ y(z) = 1 = 2" Σ amzm m=0 with ao #0 and fixed n. Solve equation (4) by identifying the coefficients am in the above power series. You should be able to find all coefficients as a function of ao. (Hint: substitute the ansatz into the equation and equate coefficients belonging to the same powers of z, then solve the resulting recurrence relation for the coefficients). 11 (2)01

Consider the second order linear differential equation z²y″(z) + zy′(z) + (z² − n²)y(z) = 0 - (4) for any fixed n = {0, 1, 2, ...}. a Is zo = 0 an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. b. Consider a power series solution around zo = 0 of the form ∞ y(z) = 1 = 2" Σ amzm m=0 with ao #0 and fixed n. Solve equation (4) by identifying the coefficients am in the above power series. You should be able to find all coefficients as a function of ao. (Hint: substitute the ansatz into the equation and equate coefficients belonging to the same powers of z, then solve the resulting recurrence relation for the coefficients). 11 (2)01

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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help me with a and b please

Consider the second order linear differential equation for any fixed n = {0, 1, 2, ...}. a Is zo = z²y" (z) + zy' (z) + (z² — n²)y(z) = 0 (4) O an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. b. Consider a power series solution around zo = 0 of the form ∞ y(z) = znΣamzm. m=0 with ao 0 and fixed n. Solve equation (4) by identifying the coefficients series. You should be able to find all coefficients as a function of ao. Am in the above power (Hint: substitute the ansatz into the equation and equate coefficients belonging to the same powers of z, then solve the resulting recurrence relation for the coefficients). C With the additional information that y(n) (0) = 2¯n, find the value for ao and write down the solution y(z) you obtain. (For fun: Does this expression look familiar?)
Transcribed Image Text:Consider the second order linear differential equation for any fixed n = {0, 1, 2, ...}. a Is zo = z²y" (z) + zy' (z) + (z² — n²)y(z) = 0 (4) O an ordinary point, a regular singular point, or an irregular singular point of the ODE? Justify your answer. b. Consider a power series solution around zo = 0 of the form ∞ y(z) = znΣamzm. m=0 with ao 0 and fixed n. Solve equation (4) by identifying the coefficients series. You should be able to find all coefficients as a function of ao. Am in the above power (Hint: substitute the ansatz into the equation and equate coefficients belonging to the same powers of z, then solve the resulting recurrence relation for the coefficients). C With the additional information that y(n) (0) = 2¯n, find the value for ao and write down the solution y(z) you obtain. (For fun: Does this expression look familiar?)
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hi, can you kindly write a detailed process about how do you get from the expression of a_(m+2) to the expression of a_(2k+1) = 0 and the expression for a_(2k) please? i am not sure how do you get to the final expression you have for a_(2k)

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