Use power series to solve the initial-value problem $$(x^2-1)y'' + 10 xy' +8 y = 0 \;, \qquad y(0) = 1\;, \qquad y'(0) = 0\;.$$

Answer: $\displaystyle y = \sum_{n=0}^\infty$ $x^{2n}$ $\displaystyle + \sum_{n=0}^\infty$ $x^{2n+1}$ .