information:classification
Differential equation definitions

The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the solution (s, c) of the system c ′ (x) = s (x), s ′ (x) = c (x), {\displaystyle {\begin{aligned}c'(x)&=s(x),\\s'(x)&=c(x),\\\end{aligned}}} with the initial conditions s (0) = 0, c (0) = 1. {\displaystyle s(0)=0,c(0)=1.} The initial conditions make the solution unique; without them any pair of functions (a e x + b e − x, a e x − b e − x) {\displaystyle (ae^{x}+be^{-x},ae^{x}-be^{-x})} would be a solution.

sinh(x) and cosh(x) are also the unique solution of the equation f ″(x) = f (x),such that f (0) = 1, f ′(0) = 0 for the hyperbolic cosine, and f (0) = 0, f ′(0) = 1 for the hyperbolic sine.