Given y'' - x^2y' + xy = 0 about (0,0)
a. find two solutions of the given differential equation.
b. solve the ivp if y(0) = 2, and y'(0) = 5
i am having trouble with the re-indexing and solving for ck's

By "ck" I'm assuming you're referring to the series solution's coefficients, i.e. .

Differentiating gives




Substituting into the ODE yields




By extracting the first term () of the third series, you can consolidate the second and third series:



Next, since the first series starts with a constant term while the other two start with a quadratic term, remove the first two terms:



Now shift the index so that the series begins at the same starting index by replacing with .



So now the series can be joined into one:



which means the ODE is equivalent to



It follows that and . Using this, you solve the recurrence



with the initial values and .

Because , it follows that ; in general, for all .

Also, because the RHS of the recurrence vanishes when , it follows that ; or generally, for , leaving you with only one not-necessarily-zero term of .

So the hard part is finding a solution for when is a multiple of 3. By finding the first few of these terms, you'll start to notice a pattern. Starting with , you find






and so on, following a general pattern of




valid for .

So the series solution is









So the two solutions are and .

To solve the IVP, notice that when , you have




so the solution to the IVP is the same as that to the general solution with the unknowns replaced accordingly.