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Pb-66.4

(a) Use the Frobenius method to find the power series solution for the differential equation (1) for a fixed N. (b) Identify a finite series solution and an infinite series solution, if they exist. (c) Is it possible to find a closed form expression for the infinite series solution?

uP''(u)-P'(u)+NP(u)			(1)

(a) The method of Frobenius applies to the power series solution of a differential equation about a regular singular point. In equations 2 and 3, the point x0=0 is proved to be a regular singular point because the limit exists.

lim(x→0)⁡〖x(-1)=0〗				(2)

lim(x→0)⁡〖x2(N/x)=0〗				(3)

Using the method of Frobenius, the power series solution (equation 4) and its first and second derivatives (equations 5 and 6, respectively) are used in the differential equation, equation 7.

P(u)=∑undefined∞〖cn 〖(x-x0)〗(n+r) 〗			(4)

P'(u)=∑undefined∞〖cn (n+r)〖(x-x0)〗(n+r-1) 〗			(5)

P(u)=∑undefined∞〖cn (n+r)(n+r-1)〖(x-x0)〗(n+r-2) 〗			(6)

∑undefined∞〖cn (n+r)(n+r-1)〖(x)〗(n+r-1) 〗-∑undefined∞〖cn (n+r)〖(x)〗(n+r) 〗+N∑undefined∞〖cn 〖(x)〗(n+r) 〗=0	(7)

Redefining the dummy variable, n, in the first term of equation 7 so that the exponent is n+r in all three summations, equation 7 can be rewritten as equation 8.

∑undefined∞〖c(n+1) (n+r+1)(n+r)〖(x)〗(n+r) 〗-∑undefined∞〖cn (n+r)〖(x)〗(n+r) 〗+N∑undefined∞〖cn 〖(x)〗(n+r) 〗=0	(8)

Then by combining all of the summations within the common range, n=0,1,…,infinity, equation 9 is defined.

c0 (r)(r-1) x(r-1)+∑undefined∞〖[c(n+1) (n+r+1)(n+r)+cn (N-r-n)] 〖(x)〗(n+r) 〗=0	(9)

Therefore, since c0 is not zero and the solution must be valid for all x greater than zero, the indicial equation is presented in equation 10 and the recurrence relationship in equation 11.

r(r-1)+0r+0=0			(10)

c(n+1)=(cn (n+r-N))/(n+r+1)(n+r) 	(11)

Using the first exponent of the differential equation, r=1, and redefining the dummy variable such that cn is a function of cn-1, an expression of the coefficients of the series solution is equation 12 for n equal to or greater than 1.

cn=(c(n-1) (n-N))/(n+1)(n) 		(12)

Rewriting the equation for the coefficients in terms of c0, equation 13 is a closed form expression of the coefficients of the series solution where (1-N)n is the pochhammer function of 1-N to the n and n is greater than or equal to 1.

cn=(c0 〖(1-N)〗n)/(n+1)!(n)! (13)

(b) Substituting the closed form expression of the coefficients back into the power series solution results in equation 14. Upon examination, the terms following the Nth term all contain the factor n-N=0; therefore, this is a closed form finite power series solution of order N.

P(u)=c0 x+∑undefined∞〖(c0 〖(1-N)〗n)/(n+1)!(n)! 〖(x)〗(n+1) 〗			(14)

(c) Using the second exponent of the differential equation, r=0, the coefficients are infinite, as expected at the regular, singular point. A second linearly independent solution may be determined using the method of reduction of order.

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MaintenanceBot (discuss • contribs) 02:50, 26 January 2014 (UTC)

EGM6611Homework1
Your EGM6611Homework1 page has been moved to User:Bartlete2/EGM6611Homework1. Many students might have an EGM6611Homework1 file, so we save them under your user space by putting User:Bartlete2/ in front. Let me know if you have any questions. -- Dave Braunschweig (discuss • contribs) 13:58, 18 February 2014 (UTC)

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MaintenanceBot (discuss • contribs) 18:57, 13 April 2014 (UTC)

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MaintenanceBot (discuss • contribs) 16:28, 28 April 2014 (UTC)