Abstract
For Euler’s differential equation of order n, a theorem is presented to give n solutions, by modifying a theorem given in a recent paper of the present authors in J. Adv. Math. Comput. Sci. 2018; 28(3): 1–15, and then the corresponding theorem in distribution theory is given. The latter theorem is compared with recent studies on Euler’s differential equation in distribution theory. A supplementary argument is provided on the solutions expressed by nonregular distributions, on the basis of nonstandard analysis and Laplace transform.
Highlights
In the preceding papers [1,2], linear differential equations of order n ∈ Z>0, with polynomial coefficients, are studied
A supplementary argument is provided on the solutions expressed by nonregular distributions, on the basis of nonstandard analysis and Laplace transform
In recent papers [5,6,7], the solution of the equation in distribution theory, which corresponds to Equation (7), is discussed
Summary
In the preceding papers [1,2], linear differential equations of order n ∈ Z>0 , with polynomial coefficients, are studied. In Reference [1,2], discussions are focused on the solution of Equation (3) with two blocks of classified terms. In recent papers [5,6,7], the solution of the equation in distribution theory, which corresponds to Equation (7), is discussed. In the Appendix A, a theorem is presented to show that there exist n and only n complementary solutions of a linear differential equation of order n, with constant coefficients, in terms of distribution theory. It guarantees the corresponding theorem on (7).
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