Consider the Second-Order Differential Equation $\neq$ 0
Assuming That A₀ = 1, One Solution of the of Frobenius

Multiple Choice

Consider the second-order differential equation .
Suppose the method of Frobenius is used to determine a power series solution of this equation. The indicial equation has r = 0 as a double root. So, one of the solutions can be represented as the power series Assume a₀ $\neq$ 0.
Assuming that a₀ = 1, one solution of the given differential equation is
Differentiating as needed, which of these relationships is correct?

A) $2 y^{\prime}(x) -a_{1}^{*}-128 a_{2}^{*} x+\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}-128 a_{n-2}^{*}\right) x^{n-1}=0$
B) $2 y^{\prime}(x) +a_{1}^{*}+128 a_{2}^{*} x+\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}+128 a_{n-2}^{*}\right) x^{n-1}=0$
C) $2 y^{\prime}(x) +a_{1}^{*}-128 a_{2}^{*} x-\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}+128 a_{n-2}^{*}\right) x^{n-1}=0$
D) $2 y^{\prime}(x) +a_{1}^{*}+128 a_{2}^{*} x+\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}-128 a_{n-2}^{*}\right) x^{n-1}=0$