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**Lancet** I need to solve the following ODE using Power Series:

$\displaystyle xy'' + y' + xy = 0$

Now, when I do it, this is what happens:

$\displaystyle y = \sum_{n=0}^{\infty} a_n x^n$

$\displaystyle y' = \sum_{n=1}^{\infty} na_n x^{n - 1}$

$\displaystyle y'' = \sum_{n=2}^{\infty} n(n - 1) a_n x^{n - 2}$

$\displaystyle x \sum_{n=2}^{\infty} n(n - 1) a_n x^{n - 2} + \sum_{n=1}^{\infty} na_n x^{n - 1} + x \sum_{n=0}^{\infty} a_n x^n = 0$

$\displaystyle \sum_{n=2}^{\infty} n(n - 1) a_n x^{n - 1} + \sum_{n=1}^{\infty} na_n x^{n - 1} + \sum_{n=0}^{\infty} a_n x^{n + 1} = 0$

Respectively:

$\displaystyle k = n - 1$

$\displaystyle k = n - 1$

$\displaystyle k = n + 1$

$\displaystyle \sum_{k=1}^{\infty} k(k + 1) a_{k + 1} x^k + \sum_{k=0}^{\infty} (k + 1)a_{k + 1} x^k + \sum_{k=1}^{\infty} a_{k - 1} x^k = 0$

$\displaystyle \sum_{k=1}^{\infty} k(k + 1) a_{k + 1} x^k + \sum_{k=1}^{\infty} (k + 1)a_{k + 1} x^k + \sum_{k=1}^{\infty} a_{k - 1} x^k + a_1 = 0$

$\displaystyle \sum_{k=1}^{\infty} [x^k[(k + 1)ka_{k + 1} + (k + 1)a_{k + 1} + a_{k - 1}]] + a_1 = 0$

$\displaystyle a_1 = 0$

$\displaystyle (k + 1)ka_{k + 1} + (k + 1)a_{k + 1} + a_{k - 1} = 0$

$\displaystyle a_{k + 1} = \frac{-a_{k - 1}}{(k + 1)k + (k + 1)}$

$\displaystyle y = a_0(1 - \frac{1}{4}x^2 + \frac{1}{64}x^4 - \frac{1}{2304}x^6 + ...)$

However, Wolfram Alpha says the solution is:

$\displaystyle y = C_1 J_0 + C_2 Y_0$

...where $\displaystyle J_n$ and $\displaystyle Y_n$ are Bessel Functions

So, did I make a mistake? And if so, where?