6: Series Solutions
We consider the homogeneous linear second-order differential equation for \(y = y(x)\): \[\label{eq:1}P(x)y''+Q(x)y'+R(x)y=0,\] where \(P(x)\), \(Q(x)\) and \(R(x)\) are polynomials or convergent power series around \(x = x_0\), with no common polynomial factors that could be divided out. The value \(x = x_0\) is called an ordinary point of \(\eqref{eq:1}\) if \(P(x_0)\neq 0\), and is called a singular point if \(P(x_0) = 0\). Singular points will later be further classified as regular singular points and irregular singular points . Our goal is to find two independent solutions of \(\eqref{eq:1}\).