The time domain solution.to the fast subsystem of the descriptor variable system of the form Ex = Ax + Bu with E singular is given by Laplace inverse transformation. A new and significant feature of the solution is that it contains impulse terms excited by the values of sufficiently smooth input u and its derivatives at the initial time point. Based on the solution, the notions of consistent initial conditions, classical solutions and impulse-free solutions are discussed perfectly. The impulse controllability is discussed also which admits the same interpretation as the controllability at infinity in frequency domain.