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When the non-homogeneous term is expressed as an exponential function, the ERF method or the undetermined coefficients method can be used to find a particular solution. If non-homogeneous terms can not be transformed to complex exponential function, then the Lagrange method of variation of parameters can be used to find solutions.
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
The following is a list of integrals of exponential functions. For a complete list of integral functions, ... and Γ(x,y) is the upper incomplete gamma function.
The definition of e x as the exponential function allows defining b x for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e x means that one has = () = for every b > 0.
Spaces within a formula must be directly managed (for example by including explicit hair or thin spaces). Variable names must be italicized explicitly, and superscripts and subscripts must use an explicit tag or template. Except for short formulas, the source of a formula typically has more markup overhead and can be difficult to read.
However, due to the multivalued nature of complex power functions for non-integer exponents, one must be careful to specify the branch of the complex logarithm being used. In addition, no matter which branch is used, if c {\displaystyle c} is not a positive integer, then the function is not differentiable at 0.
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Differentiating by x the above formula n times, then setting x = b gives: ()! = and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk.