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I've been having trouble answering these two ODE problems. Hopefully someone can help me out.

1. Compute the solution of y"' - xy' = 0 which satisfies y(0) = 1, y'(0) = 0, and y"(0) = 0.

I've tried using the power series expansion for y and substituting it in and getting the recurrence relation, but when I substitute back the initial conditions I'm getting the two series cancelling out which I don't think is right.

2. Solve the initial value problem 3y" - y' + (x+1)y = 1 with y(0) = y'(0) = 0

For this one, I know you have to compute both the power series expansion as well as a particular solution through substitution to be able to apply the initial conditions, but I'm seriously stuck in even thinking about what to substitute, let alone how to tackle this. I tried to start developing out the recurrence relationship, but it got really messy and I don't think I'm doing it right.

Thanks a lot to whoever can help, I really appreciate it!