Power Series Solutions: Definitions and Theorems

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Section 7.3 Power Series Solutions: Definitions and Theorems

In this section, we will briefly discuss the theorems that state when a second order linear ODE has power series solutions.

First, write the ode in the special form

\begin{equation} y^{\prime\prime}+p(z) y^{\prime}+q(z) y=0\text{.}\tag{7.3.1} \end{equation}

Note the coefficient one in front of the second derivative. Now exampine the functions \(p(z)\) and \(q(z)\text{,}\) thought of as functions of the complex variable \(z\text{.}\)

Definition 9. Regular Point of an ODE.

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.)

Theorem 10.

If the coefficients \(p(z)\) and \(q(z)\) are analytic at a point \(z_0\text{,}\) then a power series solution of the differential equation (7.3.1), expanded around the point \(z_0\) exists, and furthermore, the radius of convergence for the series extends at least as far as the nearest singularity (point of non-analyticity) of \(p(z)\) or \(q(z)\) in the complex plane. Usually, there will be two such power series solutions, but sometimes the second solution will be a power series times a logarithm.

Definition 11. Regular Singularity of an ODE.

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 12.

Theorem: If \(z_0\) is a regular singular point then equation (7.3.1) 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 you can look it up online or in a more comprehensive mathematical methods text, if necessary.