linear first order DEsWe can explore what kinds of integrating factors we get for coefficients of the dependent variable which are power functions of the independent variable.For fun, consider RHS functions which are also powers. The resulting integrals of power functions against exponential functions leads to some special "Whittaker" functions 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Okay, that is a bit wierd. [Whittaker was a famous mathematician! His special function solves a certain class of DEs.]
If we try it with concrete exponents, we get another special function.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Suppose we lef the RHS be an exponential function 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Maple cannot do the integral of exponentials of polynomials in the independent variable.Let's get more concrete and ignore the RHS.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This always leads to an integrating factor which is an exponential function of a power, even for a constant coefficient when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8.
This result holds for all powers except 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 but if 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 get instead an integrating factor which is instead a power of the independent variable because of the logarithmic integral.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=