Recursive Relation:

We start with the given differential equation: $(1 - x^2)y'' - 3xy' - y = 0$

Using the power series solution $y = \sum_{n=0}^\infty a_nx^n$, the recursive relation was derived as: $a_{n+2} = \frac{(3n + 1)a_n}{(n+2)(n+1)}, \quad n \geq 0$

Coefficients for Even and Odd Terms:

1. Even coefficients ($a_0, a_2, a_4, \dots$): Using the recursive formula:

   • $a_2 = \frac{(3 \cdot 0 + 1)a_0}{(2)(1)} = \frac{a_0}{2}$
   • $a_4 = \frac{(3 \cdot 2 + 1)a_2}{(4)(3)} = \frac{7a_2}{12} = \frac{7}{12} \cdot \frac{a_0}{2} = \frac{7a_0}{24}$
   • $a_6 = \frac{(3 \cdot 4 + 1)a_4}{(6)(5)} = \frac{13a_4}{30} = \frac{13}{30} \cdot \frac{7a_0}{24} = \frac{91a_0}{720}$

   Thus: $a_2 = \frac{a_0}{2}, \quad a_4 = \frac{7a_0}{24}, \quad a_6 = \frac{91a_0}{720}, \dots$

2. Odd coefficients ($a_1, a_3, a_5, \dots$): Using the recursive formula:

   • $a_3 = \frac{(3 \cdot 1 + 1)a_1}{(3)(2)} = \frac{4a_1}{6} = \frac{2a_1}{3}$
   • $a_5 = \frac{(3 \cdot 3 + 1)a_3}{(5)(4)} = \frac{10a_3}{20} = \frac{10}{20} \cdot \frac{2a_1}{3} = \frac{a_1}{3}$
   • $a_7 = \frac{(3 \cdot 5 + 1)a_5}{(7)(6)} = \frac{16a_5}{42} = \frac{16}{42} \cdot \frac{a_1}{3} = \frac{8a_1}{63}$

   Thus: $a_3 = \frac{2a_1}{3}, \quad a_5 = \frac{a_1}{3}, \quad a_7 = \frac{8a_1}{63}, \dots$

Final Results:

1. Recursive Relation: $a_{n+2} = \frac{(3n + 1)a_n}{(n+2)(n+1)}, \quad n \geq 0$

2. Specific Coefficients:

   • $a_2 = \frac{a_0}{2}, \quad a_4 = \frac{7a_0}{24}, \quad a_6 = \frac{91a_0}{720}, \dots$
   • $a_3 = \frac{2a_1}{3}, \quad a_5 = \frac{a_1}{3}, \quad a_7 = \frac{8a_1}{63}, \dots$

Would you like to compute additional terms or verify the solution further?

Related Questions:

1. How do the even and odd series separately satisfy the original differential equation?
2. How do the even and odd solutions combine into the general solution?
3. Can these series solutions be summed into closed-form expressions for $y_1(x)$ and $y_2(x)$?
4. What happens if different initial conditions ($a_0, a_1$) are chosen?
5. How does the series convergence behave near the singular points ($x = \pm 1$)?

Tip:

When calculating higher-order coefficients, track patterns in terms of factorials or ratios—they often simplify recursive series.