Abstract
We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out around the regular singular points $x=\pm1$ of the Legendre differential equations. The solutions $y_{n}(x)$ of the Legendre equation are predicted to be oscillatory in $|x| < 1$ for $n\geq3$ and nonoscillatory outside of that interval for all values of n. In contrast, the solutions $y_{n}^{m}(x)$ of the associated Legendre equation are predicted to be oscillatory for $n\geq3$ and $m\leq n-2$ only in smaller subintervals $|x| < x_{*} < 1$ , the sizes of which are determined by n and m. Numerical integrations confirm that such subintervals are distinctly smaller than $(-1, +1)$ .
Highlights
It is well known (e.g., [ ], Section . ; [ ], Section XI. ) that all ordinary second-order linear homogeneous differential equations can be written in the general form y + b(x)y + c(x)y =, ( )and that this form can be transformed to the canonical form u + q(x)u =, where the primes denote derivatives with respect to the independent variable x, q =–(b + b c)/, and y(x) = u(x) exp(–b(x) dx)
We described a program for deducing the precise intervals of oscillations in the solutions of the general form ( ), and we presented a variety of examples in which this procedure was successful in predicting oscillatory behavior by examining the coefficient q(x) alone
The rest of the equations were transformed to a form with constant damping, a quantity that clearly opposes any oscillatory tendencies in the solutions
Summary
Using the even symmetry of the canonical form ( ) with q(x) given by equation ( ), we conclude that the solutions of the Legendre equation can oscillate only in the interval We choose again c = and k = in equation ( ), in which case the transformation ( ) is applicable and the criterion for oscillatory solutions in x ∈ [ , +∞) is given by inequality ( ).
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