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