Additional solutions of a weighted heat equation. (English) Zbl 1235.34003
Consider the differential equation
\[
{d\over dx}\Biggl[(1- x^2){dv\over dx}\Biggr]= -\lambda^2 v.\tag{\(*\)}
\]
In case \(\lambda^2= n(n+1)\), \((*)\) represents the Legendre’s differential equation. Using this fact, the author describes a method to calculate infinitely many solutions to \((*)\).
Reviewer: Klaus R. Schneider (Berlin)
MSC:
34A05 | Explicit solutions, first integrals of ordinary differential equations |
34A34 | Nonlinear ordinary differential equations and systems |
References:
[1] | Rainville, E. D., Special Functions (1960), The Macmillan Company: The Macmillan Company New York · Zbl 0050.07401 |
[2] | G.A. Bear, Legendre functions as solutions to the inhomogeneous heat equation, A Thesis in Mathematics, 1995.; G.A. Bear, Legendre functions as solutions to the inhomogeneous heat equation, A Thesis in Mathematics, 1995. |
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