Frobenius, reduction of order

Find the first 4 nonzero terms in a Frobenius series solution of the given differential equation. Then use the reduction of order technique to find the logarithmic term and the first 3 nonzero terms in a second linearly independent solution.

__My solution__:

and

and

For ,

For ,

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The answer given in my textbook is:

Has the book got wrong?

Re: Frobenius, reduction of order

No. Since the characteristic values, 2 and -2, differ by an integer, you cannot get two series. That was why the problem said "use the reduction of order technique to find the logarithmic term and the first 3 nonzero terms in a second linearly independent solution."

Re: Frobenius, reduction of order

Quote:

Originally Posted by

**HallsofIvy** No. Since the characteristic values, 2 and -2, differ by an integer, you cannot get two series. That was why the problem said "use the reduction of order technique to find the logarithmic term and the first 3 nonzero terms in a second linearly independent solution."

Even when the exponents of the DE differ by an integer, there *can* be 2 linearly independent series solutions; it's just that only the series solution for the larger exponent is guaranteed.

BTW, are you implying that my answer is wrong? I've verified that *my* satisfies the given DE.