Solution of xy'' + y' + xy = 0 using series

2010-01-29T20:25:42Z

Hello,

I've tried to solve the differential equation xy'' + y' + xy = 0 using series. First, I assumed the solution:

$y=a_0+a_1x+a_2x^2+a_3x^3+...$

And from that I tried to solve it by finding out the equality between the coefficients once I substituted my solution in the differential equation. After that I get that the solution is:

$y=a_0(1-\frac{x^2}{2^2}+\frac{x^4}{2^24^2}-\frac{x^6}{2^24^46^2}+...)$

I've just started to learn about this so I'm probably missing something, but the thing is that the answer should be the Bessel function of order 0. I plotted both my solution and y = J₀(x) and they appear to be almost the same.

In case my solution is wrong, why is it? Or is there a way to work my solution to make it the Bessel function of order 0?

Thanks.

User eduardo grajeda - MathOverflow

Solution of xy'' + y' + xy = 0 using series