Section 9.2: 11, 13, 15, 17

Also:

1. The equation $y'' - 2ty' + \lambda y = 0$, where $\lambda$ is a constant, is known as Hermite's equation.
   1. Find the first five nonzero terms of two linearly independent series solutions. What is the interval of convergence?
   2. Show that Hermite's equation has a polynomial solution of degree $n$ when $\lambda = 2n$. What is this polynomial when $\lambda = 8$?
2. The equation $(1 - t^2)y'' - 2ty' + \alpha^2 y = 0$, where $\alpha$ is a constant, is known as Tchebycheff's equation.
   1. Find the first five nonzero terms of two linearly independent series solutions. What is the interval of convergence?
   2. Show that Tchebycheff's equation has a polynomial solution of degree $n$ when $\alpha = n$. What is this polynomial when $\alpha = 5$?