The point x = 0 is a regular singular point of the given differential equation. xy" â xy' + y = 0 Show that the indicial roots r of the singularity differ by an integer. (List the indicial roots below as a comma-separated list.) Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) in Section 6.3 e-SP(x) dx Y2(x) = V1(x) dx (23) O y = Cyx + C2(x Inx = 1+*+ + where necessary and a CAS, if instructed, to find a second solution. Form the general solution on (0,00). *3 2 12 72 X3 O y = Cz* + cz(x In x â 1 + + 2 6 24 + + + 72 O y = Cx In x + Cz(-1 +x+*24 (+..) O v = (x + cz(xet - 1) O v=C« In x+C2(-1+x+*+ ...)

The biologist has a second culture to examine. she knows that the population of the culture doubles every 15 minutes. after 1 hour and 15 minutes, her assistant found that 80,000 bacteria were present. a) what was the size of the initial population? b) predict the size of the culture at t=3 hours. what was the size of the population at 40 minutes? explain and justify your answers. c) create a graph of the population as a function of time. find an equation that can be used to predict the size of the population at any time t. d) examine the rate at which the bacteria culture is growing. how fast is the culture growing after 1 hour? after 1.5 hours? after 2 hours? use a time interval of h = 0.01 hours to estimate these rates. interpret these rates in terms of the context of the problem situation. how do these three rates compare?