Use an appropriate infinite series method about x 0 to find

Solution

Here x=0 is an ordinary point. So let y = a0 + a1x+ a2x2 + .... + anxn + .... Now putting it in the given differential equation we have, (2.1.a2 + 3.2.a3x + 4.3.a4x2 + .... + n(n-1)anxn-2 + ....) - x2(a1 + 2a2x + 3a3x2 + ..... ) + x(a0 + a1x+ a2x2 + .... + anxn + ....) = 0 Equating coefficients of various powers of x to 0, Coefficient of x0 : a2 = 0. Coefficient of x1 : 3.2a3 + a0 = 0 Do this for a few powers of x (generally upto x3 would do). Now we get the general relation by equating coefficient of xn to 0. We get: an+2 =[ (n-2)*an-1]/[(n+1)(n+2)] Putting n=4,5,6... we get the desired series solution!