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.

15%

step-by-step explanation:

use the table and count the number of guesses that has two correct answers.

the following are two correct answers

10010, 01100, 11000

a total of 3 numbers having 2 correct guesses out of 20 questions.

probability of getting 2 correct guesses = the number of questions getting 2 correct answers / total number of questions.

= 3/20

to convert it into percent multiply by 100, we get

= 3/20 * 100

= 15%

the answer is 15%

you.

answer: are you

step-by-step explanation: