Exponential approach

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In mathematics, the exponential approach is understood to be an approach to solving a linear differential equation with constant coefficients, the inhomogeneity of which has an exponential structure. The idea is that a particulate solution of a similar shape to the inhomogeneity then also exists. Such a solution approach reduces the differential equation to a linear system of equations . The idea for this approach goes back to Leonhard Euler .


A linear differential equation is given

with constant coefficients , wherein the inhomogeneity is the structure

owns. Also denote the zero order of with respect to the characteristic polynomial of the associated homogeneous equation

Then there is a special solution of the form


Consider the linear differential equation

Now is the first order zero of the polynomial . So according to the above theorem there is a special solution of the shape



is obtained from the differential equation

Comparison of coefficients provides the determining equations

which and implies. So is

a special solution to the above inhomogeneous differential equation.


  • Harro Heuser: Textbook of Analysis Part 1 . 5th edition. Teubner-Verlag 1988, ISBN 3-519-42221-2 , pp. 413-428.