
# 7: Power series methods

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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 5th and 9th degree.