The Solutions of the Differential Equation for Specific Excitations

  • Gustav Doetsch


When the differential equation (15.13) describes some physical system, then the solution 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 y0, y0, …, y0 (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
$${y_0}\, = \,{y'_0}\, = \, \ldots \, = \,y_0^{\left( {n - 1} \right)}\, = \,0$$
; that is, the system is assumed to be initially at rest. Thus we find the corresponding solution in the image space:
$$Y\left( s \right)\, = \,G\left( s \right)F\left( s \right)$$
and, consequently, in the original space:
$$y\left( t \right)\, = \,g\left( t \right) * f\left( t \right).$$


Simple Root Imaginary Axis Image Function Step Response Original Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Gustav Doetsch
    • 1
  1. 1.Emeritus of MathematicsUniversity of FreiburgFreiburg i. Br.Germany

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