Solve this differential equation using power series and indicial roots about (0,0):

3xy'' + 6y' + y = 0

Let , so that




Substituting into the ODE gives




The first series starts with a linear term, while the other two start with a constant. Extract the first term from each of the latter two series:




Finally, to get the series to start at the same index, shift the index of the first two series by replacing with . Then the ODE becomes



which can be consolidated to get




You're fixing the solution so that it contains the origin, which means



which in turn means . With the given recurrence, it follows that for all , so the solution would be . This is to be expected, since is clearly a singular point for the ODE.