Consider the differential equationthat appeared in an advertisement for a symbolic algebra...

Consider the differential equation

that appeared in an advertisement for a symbolic algebra program in the March 1984 issue of the American Mathematical Monthly. (a) Show that x = 0 is a regular singular point with exponents r1 = 1 and r2 = 0. (b) It follows from Theorem 1 that this differential equation has a power series solution of the form

Substitute this series (with C1 = 1) in the differential equation to show that C2 = −2, C3 = 3, and

for n ≧ 2. (c) Use the recurrence relation in part (b) to prove by induction that Hence deduce (using the geometric series) that

for 0 < x < 1.