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For some equations that we cannot solve explicitly, we may try to expand the solutions into power series. This chapter starts with a reminder about power series and explains why the natural domain to discuss power series is the complex plane. We illustrate solution by power series for second-order normalized linear equations whose coefficients can be expanded into power series around some point (i.e., are analytic there) and show some techniques to manipulate the series. Section 7.3 introduces the case of regular-singular points. Sections 7.4 and 7.5 discuss the cases of equal indices and indices that differ by an integer. The last section is devoted to the Bessel equation.

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Solution of Differential Equations by Power Series

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  • 发表日期：

    2025年

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  • Uri Elias

摘要

For some equations that we cannot solve explicitly, we may try to expand the solutions into power series. This chapter starts with a reminder about power series and explains why the natural domain to discuss power series is the complex plane. We illustrate solution by power series for second-order normalized linear equations whose coefficients can be expanded into power series around some point (i.e., are analytic there) and show some techniques to manipulate the series. Section 7.3 introduces the case of regular-singular points. Sections 7.4 and 7.5 discuss the cases of equal indices and indices that differ by an integer. The last section is devoted to the Bessel equation.