Editing Exponential function (section)

==Differential equations==
{{main|Linear differential equation}}
Exponential functions occur very often in solutions of [[differential equation]]s.

The exponential functions can be defined as solutions of [[differential equation]]s. Indeed, the exponential function is a solution of the simplest possible differential equation, namely {{tmath|1=y'=y}}. Every other exponential function, of the form {{tmath|1=y=ab^x}}, is a solution of the differential equation {{tmath|1=y'=ky}}, and every solution of this differential equation has this form.

The solutions of an equation of the form 
<math display=block>y'+ky=f(x)</math> 
involve exponential functions in a more sophisticated way, since they have the form
<math display=block>y=ce^{-kx}+e^{-kx}\int f(x)e^{kx}dx,</math> 
where {{tmath|c}} is an arbitrary constant and the integral denotes any [[antiderivative]] of its argument. 

More generally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential functions and, when they are not homogeneous, antiderivatives. This holds true also for systems of linear differential equations with constant coefficients.