The second order equation 5xy" + 13y' + xy = 0 has a regular singular point at a = 0, and has series solutions of the form n=0 y = Cnxn+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 ) x-+( (2) From the indicial equation we can solve the indicial roots 0,-8/5 0 (3) Let r be the smaller indicial root. Then -8/5 Let c = 1. From the recurrence relation, we have a solution y = x+0 1+2+ +1+-1/2^2 Cn = (4) Let be the larger indicial root. Then = 0 Cn = )x + n=2 Cn = and the recurrence relation becomes x+3+ = 0 Cn-2 for n = 2, 3, ... (enter your results as a comma separated list). For any one of the indicial roots, we have c = and the recurrence relation becomes Let Co = 1. From the recurrence relation, we have another solution y2 = x+ 0 2+1+-1/(2*18) x+2+ x+3+ (5) The general solution is given by y = Ay+ By with arbitrary constants A, B. )cn + Cn-2 for n = 2, 3, ... +4+ Cn-2 for n = 2, 3, ... +4+ x+5+ Cn-2)x+r-10 +5+ 271+6 x2+6 + +...