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Section 9.10 Power Series Solutions: Theorem

In this section, we will briefly discuss the theorem that states when a second order linear ode has power series solutions.

First, write the ode in the form:

\begin{equation} y^{\prime\prime}+p(z) y^{\prime}+q(z) y=0\label{diffeq}\tag{9.10.1} \end{equation}

and look at the functions \(p(z)\) and \(q(z)\) has function of the complex variable \(z\text{.}\) If \(p(z)\) and \(q(z)\) are analytic at a point \(z=z_0\text{,}\) the \(z_0\) is said to be a regular point of the differential equation. (The word analytic is a technical term for a complex-valued function which is (complex) differentiable at the point. You can learn more about this concept online. But for practical purposes, it means that the function does not blow up at the point \(z_0\text{,}\) nor is it otherwise badly behaved, e.g. the origin of a square root.)

Definition 9.10.2.

If \((z-z_0) p(z)\) and \((z-z_0)^2 q(z)\) are analytic, then the point \(z_0\) is called a regular singular point or regular singularity. Theorem: If \(z_0\) is a regular singular point then equation (9.10.1) can be solved by an extension of power series methods called a Frobenius Series. The solution will consist of: (1) two Frobenius series, or (2) one Frobenius series \(y_1(z-z_0)\) and a second solution \(y_2(z-z_0)=y_1(z-z_0)\ln(z-z_0)+y_0(z-z_0)\text{,}\) where \(y_0(z-z_0)\) is a second Frobenius series. We will not discuss this method further here, but again, you can look it up online or in a more comprehensive mathematical methods text, if necessary.