The second order equation 5xy + 13y' + xy = 0 has a regular singular point...

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The second order equation 5xy" + 13y' + xy = 0 has a regular singular point at a = 0, and has series solutions of the form Σ 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 ) ²₁x²-¹+( (2) From the indicial equation we can solve the indicial roots 0,-8/5 0 (3) Let r₁ be the smaller indicial root. Then ₁ -8/5 Let c = 1. From the recurrence relation, we have a solution y₁ = x¹¹+0 1+2+ +1+-1/2^2 Cn = (4) Let ₂ be the larger indicial root. Then ₂ = 0 Cn = )₁x² + 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 c₁ = and the recurrence relation becomes Let Co = 1. From the recurrence relation, we have another solution y2 = x¹²+ 0 2+1+-1/(2*18) x₂+2+ x₂+3+ (5) The general solution is given by y = Ay₁+ By₂ with arbitrary constants A, B. )cn + Cn-2 for n = 2, 3, ... +4+ Cn-2 for n = 2, 3, ... ₂+4+ x+5+ Cn-2)x+r-10 +5+ 271+6 x2+6 + +... The second order equation 5xy" + 13y' + xy = 0 has a regular singular point at a = 0, and has series solutions of the form Σ 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 ) ²₁x²-¹+( (2) From the indicial equation we can solve the indicial roots 0,-8/5 0 (3) Let r₁ be the smaller indicial root. Then ₁ -8/5 Let c = 1. From the recurrence relation, we have a solution y₁ = x¹¹+0 1+2+ +1+-1/2^2 Cn = (4) Let ₂ be the larger indicial root. Then ₂ = 0 Cn = )₁x² + 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 c₁ = and the recurrence relation becomes Let Co = 1. From the recurrence relation, we have another solution y2 = x¹²+ 0 2+1+-1/(2*18) x₂+2+ x₂+3+ (5) The general solution is given by y = Ay₁+ By₂ with arbitrary constants A, B. )cn + Cn-2 for n = 2, 3, ... +4+ Cn-2 for n = 2, 3, ... ₂+4+ x+5+ Cn-2)x+r-10 +5+ 271+6 x2+6 + +...

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Recurrence Relation and Indicial Equation By replacing one can obtain the indicial equation 0 n0 int... View the full answer