The second order equation 3xy" + 11xy' + (1x 3) y = 0 has a regular singular point at x = 0, and has series solutions of the form - n=0 y = Cnx+r (1) Insert the formal power series into the differential equation, we derive an equation So we have the indicial equation and a recurrence relation (3) Let r be smaller indicial (2) From the indicial equation we can solve the indicial roots -3,1/3 Y = x+ Let co= 1. From the recurrence relation, we have a solution T+1 x + hen r = (4) Let be the larger indicial root. Then 1/3 ) cox + (( n=1 x2+1 x+2+ + Cn = and the recurrence rela C = Let co= 1. From the recurrence relation, we have another solution +2+ Cn = x71+3 (enter your results as a comma separated list). +... 22+3+... Y2 = x2+ (5) The general solution is given by y = Ay+By2 with arbitrary constants A, B. )en + Cn-1 for n = 1, 2, ... and the recurrence relation becomes = 0 becomes Cn-1 for n = 1, 2,... Cn-1 for n 1, 2, ... Cn-1)x+r = 0