Advanced Engineering Mathematics, Lecture 2.6: Singular points and the Frobenius method.

In the previous lecture, we saw how many 2nd ODEs have solutions that are power series. In this lecture, we learn when this is the case, which is due to a theorem of Frobenius. In particular, if the center x_0 of the power series is a so-called "ordinary point", then the ODE will have a power series solution. Otherwise, x_0 is said to be a "singular point," and there are two types of these. If x_0 is an "regular singular point", then there is a generalized power series solution, which is simply a power series times x^r, where r could be a negative number, fraction, or complex number. In this lecture, we learn what these mean, we discuss the radii of convergence of these solutions, and we do an example of an ODE that has as generalized power series solution. Finally, we discuss a number of classic ODEs that arise in physics and engineering that have power series or generalized power series solutions, including ODEs named after Legendre, Chebyshev, Hermite, Bessel, Laguerre, and Airy.

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