[MATH](1 - x^2)y'' - xy' + p^2 y = 0[/MATH]
Find and solve the indicial equation and compute the recurrence relations for two linearly independent solutions about the point [MATH]x = 1[/MATH].
I plugged the guess [MATH]y = x^s \sum_{n = 0}^\infty a_n x^n[/MATH] into the ODE and obtained [MATH](s^2 - s)a_0 x^{s - 2} + (s^2 + s)a_1 x^{s - 1} + \sum_{n = 0}^\infty (n + s + 2)(n + s + 1)a_{n + 2} x^{n + s} + (p^2 - n^2 - 2ns - s^2)a_n x^{n + s} = 0[/MATH].
Normally, when the point of interest is [MATH]x = 0[/MATH], the indicial equation would set the coefficient of [MATH]a_0 x^{f(s)}[/MATH] to [MATH]0[/MATH], in this case, [MATH]s^2 - s = 0 \implies s = 0, 1[/MATH]. Since the point of interest is [MATH]x = 1[/MATH], should I set the coefficient of [MATH]a_1 x^{g(s)}[/MATH] to [MATH]0[/MATH] instead, in this case [MATH]s^2 + s = 0 \implies s = -1, 0[/MATH]? I'm not really sure how contingent the decision to single out the [MATH]a_0[/MATH] term is. Alternatively, was the aim of the problem specifying [MATH]x = 1[/MATH] for me to make the guess [MATH]y = x^s \sum_{n = 0}^\infty a_n (x - 1)^n[/MATH] instead, since [MATH]\sum_{n = 0}^\infty a_n (x - 1)^n[/MATH] is how a Maclaurin Series [MATH]\sum_{n = 0}^\infty a_n x^n[/MATH] would be rewritten as a Taylor Series centered around [MATH]x = 1[/MATH]? How does the given point [MATH]x = 1[/MATH] change how I proceed to the recurrence relations?
Find and solve the indicial equation and compute the recurrence relations for two linearly independent solutions about the point [MATH]x = 1[/MATH].
I plugged the guess [MATH]y = x^s \sum_{n = 0}^\infty a_n x^n[/MATH] into the ODE and obtained [MATH](s^2 - s)a_0 x^{s - 2} + (s^2 + s)a_1 x^{s - 1} + \sum_{n = 0}^\infty (n + s + 2)(n + s + 1)a_{n + 2} x^{n + s} + (p^2 - n^2 - 2ns - s^2)a_n x^{n + s} = 0[/MATH].
Normally, when the point of interest is [MATH]x = 0[/MATH], the indicial equation would set the coefficient of [MATH]a_0 x^{f(s)}[/MATH] to [MATH]0[/MATH], in this case, [MATH]s^2 - s = 0 \implies s = 0, 1[/MATH]. Since the point of interest is [MATH]x = 1[/MATH], should I set the coefficient of [MATH]a_1 x^{g(s)}[/MATH] to [MATH]0[/MATH] instead, in this case [MATH]s^2 + s = 0 \implies s = -1, 0[/MATH]? I'm not really sure how contingent the decision to single out the [MATH]a_0[/MATH] term is. Alternatively, was the aim of the problem specifying [MATH]x = 1[/MATH] for me to make the guess [MATH]y = x^s \sum_{n = 0}^\infty a_n (x - 1)^n[/MATH] instead, since [MATH]\sum_{n = 0}^\infty a_n (x - 1)^n[/MATH] is how a Maclaurin Series [MATH]\sum_{n = 0}^\infty a_n x^n[/MATH] would be rewritten as a Taylor Series centered around [MATH]x = 1[/MATH]? How does the given point [MATH]x = 1[/MATH] change how I proceed to the recurrence relations?