9: Series Solutions of homogeneous linear second-order differential equations
Reference: Boyce and DiPrima, Chapter 5
We consider the homogeneous linear second-order differential equation for \(y=\) \(y(x)\) :
\[P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0, \nonumber \]
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 Equation \ref{9.1} if \(P\left(x_{0}\right) \neq 0\) , and is called a singular point if \(P\left(x_{0}\right)=0\) . Singular points can be further classified as regular singular points and irregular singular points. Here, we will only consider series expansions about ordinary points. Our goal is to find two independent solutions of Equation \ref{9.1}.