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method of undetermined coefficients, ordinary differential equations, hyper- bolic cosines and sines


Mathematics | Physical Sciences and Mathematics | Science and Mathematics Education


The method of undetermined coefficients is commonly applied to solve linear, constant coefficient, non-homogeneous ordinary differential equations when the forcing function is from a selected class of functions. Often the hyperbolic sine and cosine functions are not explicitly included in this list of functions. Through a set of guided examples, this work argues that the hyperbolic sine and cosine ought to be included in the select class of functions. Careful explanation is provided for the necessary treatment of the cases where the argument of the hyperbolic sine and/or cosine functions matches one or both of the roots of the characteristic equation of the differential equation. Finally, a generalized approach where the hyperbolic and trigonometric sine and cosine functions are written in their exponential form illustrates the connections between the exponential, trigonometric, and hyperbolic sine and cosine functions. This exploration leads to a deeper understanding of the method of undetermined coefficients and can be adapted into coursework on the undetermined coefficients topic.



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