Power series differential equation

given
a) use power series method to find general solution near x=0

i already got the recurrence equation of

i just need to know how to get the general solution

The next step is to solve the recurrence, but let's back up a bit. You should have found that the ODE in terms of the power series expansion for



which indeed gives the recurrence you found,



but in order to get anywhere with this, you need at least three initial conditions. The constant term tells you that , and substituting this into the recurrence, you find that for all .

Next, the linear term tells you that , or .

Now, if is the first term in the sequence, then by the recurrence you have





and so on, such that for all .

Finally, the quadratic term gives , or . Then by the recurrence,






and so on, such that



for all .

Now, the solution was proposed to be



so the general solution would be