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

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 .
Assume x > 0. Which of these is a form of a second solution of the given differential equation, linearly independent to (x) ?

A) $y_{2}(x) =y_{1}(x) \ln x+\sum_{n=1}^{\infty} a^{*}{ }_{n} x^{n}$
B) $y_{2}(x) =y_{1}^{\prime}(x) \ln x \mid+\sum_{n=1}^{\infty} a^{*} n^{n}$
C) $y_{2}(x) =\ln x+y_{1}(x) \cdot \sum_{n=1}^{\infty} a^{*}{ }_{n} x^{n}$
D) $y_{2}(x) =|\ln x|+y_{1}(x) \cdot \sum_{n=1}^{\infty} a^{n}{ }_{n} x^{n}$
E) $y_{2}(x) =\ln x\left(y_{1}(x) +\sum_{n=1}^{\infty} a^{n}{ }_{n} x^{n}\right)$