# 7: Power series methods

- Page ID
- 356

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- 7.1: Power Series
- Many functions can be written in terms of a power series. If we assume that a solution of a diﬀerential equation is written as a power series, then perhaps we can use a method reminiscent of undetermined coeﬃcients. That is, we will try to solve for the coefficients of the expansion . Before we can carry out this process, let us review some results and concepts about power series.

- 7.2: Series solutions of linear second order ODEs
- For linear second order homogeneous ODEs with polynomials as functions can often be solved by expanding functions around ordinary or specific points.

- 7.3: Singular Points and the Method of Frobenius
- While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.

- 7.E: Power series methods (Exercises)
- These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.

*Thumbnail: The sine function and its Taylor approximations around \(x_o=0\) of 5 ^{th} and 9^{th} degree.*

### Contributors

- Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS-0900885 and DMS-1362337.