Euler-Cauchy Undetermined Coefficients Exception
United States Service Academies
In 2013 we stood up the Army Armament Graduate School at Picatinny Arsenal in New Jersey. All PhD students take Advanced Mathematics I and II. For the past eight years, we have taught the method of undetermined coefficients (UC) as a solutions technique for linear nonhomogeneous ordinary differential equations with constant coefficients whose nonhomogeneous terms are members of a short, but important, list (Kreyzsig, 2011). For this method to work, the derivatives of the nonhomogeneous term must form a closed set (Web references, 2021). Otherwise a more involved method, variation of parameters (VoP), is employed. This eighth time teaching, we noticed an exception to the rule for the implementation of the method of undetermined coefficients. Before jumping in with the exception, we introduce four topics that are useful in this problem, and that we also wish that all undergraduates would know. The first is why like powers of the variable, or like special functions, or the real part and the imaginary parts need to separately cancel each other out in order for an equation to be true over an open interval. The second is the set of functions that regenerate themselves, phoenix-like, upon differentiation. The third is avoiding division by zero, and when you can remove a singularity by multiplying both sides of the equation by the denominator. The fourth is the power law in derivatives.
George Fischer and Laurie Florio. "Euler-Cauchy Undetermined Coefficients Exception". Mathematica Militaris, Volume 25. 2022.