TeachingDifferential equations – Series solutions of linear differential equations
Definition: A function \(f\) is called analytic at \(x=x_0\) if it has a Taylor series expansion about \(x=x_0\):
\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n\]with a positive radius of convergence.
We consider homogeneous second-order linear differential equations of the form
\[P(x)y''(x)+Q(x)y'(x)+R(x)y(x)=0\]where \(P\), \(Q\) and \(R\) are polynomials which do not have a common zero.
Definition: A point \(x_0\) is called a regular point (in the book: ordinary point) if \(P(x_0)\neq0\). Otherwise, so if \(P(x_0)=0\), it is called a singular point.
Suppose that \(x_0\) is a regular point of the differential equation. Since \(P(x)\) is continuous, it follows that there exists an open interval \(I\) containing \(x_0\) in which \(P(x)\) is never zero. In that interval \(I\) the differential equation can be divided by \(P(x)\) to obtain
\[y''(x)+p(x)y'(x)+q(x)y(x)=0,\]where \(p(x)=\displaystyle\frac{Q(x)}{P(x)}\) and \(q(x)=\displaystyle\frac{R(x)}{P(x)}\) are continuous functions on \(I\). Hence, the existence and uniqueness theorem implies that there exists a unique solution of the differential equation in the interval \(I\) that also satisfies the initial conditions \(y(x_0)=x_0\) and \(y'(x_0)=y_0'\) for arbitrary values of \(y_0\) and \(y_0'\). In that case the general solution of the differential equation can be written as a power series of the form \(y(x)=\displaystyle\sum_{n=0}^{\infty}c_n(x-x_0)^n\) with a positive radius of convergence.
On the other hand, if \(x_0\) is a singular point of the differential equation, then the existence of uniqueness theorem does not hold. However, in some cases the method of Frobenius can be applied to find solutions of the form \(y(x)=(x-x_0)^r\displaystyle\sum_{n=0}^{\infty}c_n(x-x_0)^n\) for some \(r\in\mathbb{R}\). This is called a generalized power series.
Now we consider homogeneous second-order linear differential equation which can be written in the form
\[y''(x)+p(x)y'(x)+q(x)y(x)=0.\]Definition: A point \(x_0\) is called a regular point of the differential equation if both \(p(x)\) and \(q(x)\) are analytic at \(x=x_0\). Otherwise, \(x_0\) is called a singular point of the differential equation.
Definition: If \(x_0\) is a singular point of the differential equation and both \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic at \(x=x_0\), then \(x_0\) is called a regular singular point of the differential equation. Otherwise, \(x_0\) is called an irregular singular point of the differential equation.
Last modified on May 13, 2021
