Problem #4 (20 points) - Using The Method Of Taylor
Consider the ODE
$(a + bx^2)y''(x) + cxy'(x) + \lambda y(x) = 0$
in which $a$, $b$, $c$ and $\lambda$ are constants.
a.) (10 points) Determine (in terms of $a$, $b$, $c$ and $n = 0, 1, 2, 3, ...$) how $\lambda = \lambda_n$ should be
chosen so that one solution to this ODE is a polynomial.
b.) (10 points) Determine the explicate form of the polynomial solution to this ODE for the
case when
$a = 1$, $b = -4$, $c = -4$ and $\lambda = 144$,
assuming also that this polynomials equals one when $x = 1/2.$
