Chapter 11.7, Problem 15AP

Consider the differential equation

$\left(x^2-1\right)y^{″}+\left[1-\left(a+b\right)\right]x{y}^{\prime }+aby=0$, … $\left(\mathrm{11.7.9}\right)$

where $a$ and $b$ are constants.

(a) Show that the coefficients in a series solution to Equation $\left(\mathrm{11.7.9}\right)$ centered at $x=0$ must satisfy the recurrence relation

$a_{n+2}=\frac{\left(n-a\right)\left(n-b\right)}{\left(n+2\right)\left(n+1\right)}{a}_n,n=0,1,\dots$,

and determine two linearly independent series solutions.

(b) Show that if either $a$ or $b$ is nonnegative integer, then one of the solutions obtained in (a) is a polynomial.

(c) Show that if $a$ is an odd positive integer and $b$ is an even positive integer, then both of the solutions defined in (a) are polynomials.

(d) If $a=5$ and $b=4$, determine two linearly independent polynomial solutions to Equation $\left(\mathrm{11.7.9}\right)$. Notice that in this case the radius of convergence of the solutions obtained is $R=\infty$, whereas Theorem 11.2.1 only guarantees a radius of convergence $R\ge 1$.