find two linearly independent power series solutions about the point x0 = 0 of
(x^2 ? 4)y"+ 3xy' + y = 0. state the minimum radius of convergence, and explain
why it is the minimum of radius of convergence.

Assume a solution of the form

with derivatives

Substituting into the ODE, which appears to be

gives

which gives the recurrence for the coefficients ,

There's dependency between coefficients that are 2 indices apart, so we consider 2 cases.

and so on, with the general pattern

and so on, with

Then the two independent solutions to the ODE are

and

By the ratio test, both series converge for , which also can be deduced from the fact that are singular points for this ODE.

step-by-step explanation:

the multiplicative inverse of a fraction a/b is b/a. for the multiplicative inverse of a real number, divide 1 by the number. for example, the reciprocal of 5 is one fifth 1/5 or 0.2, and the reciprocal of 0.25 is 1 divided by 0.25, or 4.