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.

d. millimeters per hour.

step-by-step explanation:

we are given that,

carlo used a rain gauge to measure the amount of water his lawn is receiving from the sprinklers.

since, according to the options, we see that,

water cannot be measured by 'miles per hour', 'meters per second' and 'grams per seconds'.

we get that, the water can be measured by 'millimeters per hour'.

hence, the appropriate rate for describing the amount of water is 'millimeters per hour'.

1 x=22.52

2: 65/39=604+39x

65x-39x=604

26x=604

x=604/26

×=23.23