Editing Hyperbolic functions (section)

=== Differential equation definitions ===

The hyperbolic functions may be defined as solutions of [[differential equation]]s: The hyperbolic sine and cosine are the solution {{math|(''s'', ''c'')}} of the system
<math display="block">\begin{align}
c'(x)&=s(x),\\
s'(x)&=c(x),\\
\end{align}
</math>
with the initial conditions <math>s(0) = 0, c(0) = 1.</math> The initial conditions make the solution unique; without them any pair of functions <math>(a e^x + b e^{-x}, a e^x - b e^{-x})</math> would be a solution.

{{math|sinh(''x'')}} and {{math|cosh(''x'')}} are also the unique solution of the equation {{math|1=''f''&thinsp;″(''x'') = ''f''&thinsp;(''x'')}},
such that {{math|1=''f''&thinsp;(0) = 1}}, {{math|1=''f''&thinsp;′(0) = 0}} for the hyperbolic cosine, and {{math|1=''f''&thinsp;(0) = 0}}, {{math|1=''f''&thinsp;′(0) = 1}} for the hyperbolic sine.