# The Solutions of the Differential Equation for Specific Excitations

Chapter

## Abstract

When the differential equation (15.13) describes some physical system, then the solution ; that is, the system is assumed to be initially at rest. Thus we find the corresponding solution in the image space: and, consequently, in the original space:

*y*(*t*) of the homogeneous equation represents the action of the system that is left undisturbed, starting from an initial situation which is defined by the initial values*y*_{0},*y*′_{0}, …,*y*_{0}^{(n−1)}. This solution is a linear combination of functions of the type*t*^{ k }*e*^{ at }; it is easy to survey. Therefore we disregard this part of the problem here, and we presume that$${y_0}\, = \,{y'_0}\, = \, \ldots \, = \,y_0^{\left( {n - 1} \right)}\, = \,0$$

(1)

$$Y\left( s \right)\, = \,G\left( s \right)F\left( s \right)$$

(2)

$$y\left( t \right)\, = \,g\left( t \right) * f\left( t \right).$$

(3)

## Keywords

Simple Root Imaginary Axis Image Function Step Response Original Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1974