Consider the Second-Order Differential Equation $x^{2} \cdot \frac{7 x(x+1)}{5 x^{2}}$

Multiple Choice

Consider the second-order differential equation: .
Why is C₀ = 0 a regular singular point?

A) The functions $x^{2} \cdot \frac{7 x(x+1) }{5 x^{2}}$ and $x \cdot\left(-\frac{7}{5 x^{2}}\right)$ both have convergent Taylor series expansions about 0 .
B) The functions $x \cdot \frac{7 x(x+1) }{5 x^{2}}$ and $x^{2} \cdot\left(-\frac{7}{5 x^{2}}\right)$ both have convergent Taylor series expansions about 0 .
C) $\lim _{x \rightarrow 0} x \cdot\left(-\frac{7}{5 x^{2}}\right) =\infty$
D) $\lim _{x \rightarrow 0} x \cdot\left(-\frac{7}{5 x^{2}}\right) \neq 0$