5. find at least the first 4 nonzero terms of the solution to (2 + 4x-2x^2)y"-12(z-1)y'-12y = 0, y(1) = 0, y'(1) =-1.

We're set up to find a series solution to the ODE centered around . Let's first replace , so that

and we'll look for a series solution centered at . We have

The ODE is then

which in terms of is

Substituting the series above into the ODE gives

so that the coefficients follow the recurrence,

Thanks to the dependency between every other coefficient, we have for all even-valued . Meanwhile,

so that

or in terms of ,

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step-by-step explanation: