Conditions for the unique solvability of the initial-value problem for linear second-order differential equations with argument deviations

2006 ◽  
Vol 9 (4) ◽  
pp. 523-535 ◽  
Author(s):  
A. M. Ronto ◽  
N. Z. Dil’na
Keyword(s):  
Differential Equations ◽  
Unique Solvability ◽  
Initial Value Problem ◽  
Second Order ◽  
Initial Value ◽  
Second Order Differential Equations

Superexponentials: A Generalization of Hyperbolic and Trigonometric Functions

2017 ◽  
Vol 3 (1) ◽  
pp. 7
Author(s):  
Alfonso F. Agnew ◽  
Brandon Gentile ◽  
John H. Mathews
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Second Order ◽  
Series Expansions ◽  
Roots Of Unity ◽  
Trigonometric Functions ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Power Series Expansions

We construct and explore the properties of a generalization of hy- perbolic and trigonometric functions we cal l superexponentials. The general ization is based on the characteristic second-order differential equations (DE) these functions satisfy, and leads to functions satisfying analogous mth order equations and having many properties analogous to the usual hyperbolic and trigonometric functions. Roots of unity play a key role in providing the periodicity resulting in various properties. We also show how these functions solve the general initial value problem for the differential equations y(n) = y, and a look at the power series expansions reveal surprisingly simple patterns that clarify the properties of the superexponentials.

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