Consider the Second-Order Differential Equation $y_{1}(x)=x^{-4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=\ln x\left(y_{1}(x)+x^{-4} \sum_{n=0}^{\infty} b_{n} x^{n}\right)$

Multiple Choice

Consider the second-order differential equation .
Which of the following is the form of a pair of linearly independent solutions of this differential equation?

A) $y_{1}(x) =x^{-4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x) =\ln x\left(y_{1}(x) +x^{-4} \sum_{n=0}^{\infty} b_{n} x^{n}\right)$
B) $y_{1}(x) =x^{4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x) =\ln x\left(y_{1}(x) +x^{4} \sum_{n=0}^{\infty} b_{n} x^{n}\right)$
C) $y_{1}(x) =x^{-4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x) =y_{1}(x) \ln x+x^{-4} \sum_{n=0}^{\infty} b_{n} x^{n}$
D) $\left.y_{1}(x) =x^{4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x) =y_{1} x\right) \ln x+x^{4} \sum_{n=0}^{\infty} b_{n} x^{n}$