User eduardo grajeda - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T04:28:35Z http://mathoverflow.net/feeds/user/3636 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13412/solution-of-xy-y-xy-0-using-series Solution of xy'' + y' + xy = 0 using series Eduardo Grajeda 2010-01-29T20:25:42Z 2010-01-29T20:25:42Z <p>Hello,</p> <p>I've tried to solve the differential equation <em>xy'' + y' + xy = 0</em> using series. First, I assumed the solution:</p> <p>$y=a_0+a_1x+a_2x^2+a_3x^3+...$</p> <p>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:</p> <p>$y=a_0(1-\frac{x^2}{2^2}+\frac{x^4}{2^24^2}-\frac{x^6}{2^24^46^2}+...)$</p> <p>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 <em>y = J<sub>0</sub>(x)</em> and they appear to be almost the same.</p> <p>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?</p> <p>Thanks.</p>