In Introduction to Power Series, we studied how functions can be represented as power series, [latex]y(x)=\displaystyle\sum_{n=0}^{\infty} a_{n}x^{n}[/latex]. We also saw that we can find series representations of the derivatives of such functions by differentiating the power series term by term. This gives [latex]y^{\prime}(x)=\displaystyle\sum_{n=1}^{\infty} na_{n}x^{n-1}[/latex] and [latex]y^{\prime\prime}(x)=\displaystyle\sum_{n=2}^{\infty} n(n-1)a_{n}x^{n-2}[/latex]. In some cases, these power series representations can be used to find solutions to differential equations.

Be aware that this subject is given only a very brief treatment in this text. Most introductory differential equations textbooks include an entire module on power series solutions. This text has only a single section on the topic, so several important issues are not addressed here, particularly issues related to existence of solutions. The examples and exercises in this section were chosen for which power solutions exist. However, it is not always the case that power solutions exist. Those of you interested in a more rigorous treatment of this topic should consult a differential equations text.