Usually, impulsive differential equations have discontinuity in the waveform where this discontinuity is represented by an impulse. A discontinuity in the waveform means that the derivative at this time point is equal to infinity.

Usually, the numerical techniques for solving ODE are based on the assumption that the waveform is continuous (No derivatives equal to infinity) at the time point you are solving for. In the case of discontinuity , the solution would depend on how quickly the integration algorithm would recover from this discontinuity.

In the case of having L-stable integration method it is expected that the integration method would recover quickly. However if the method is not L-stable you might see a lot of oscillations in the results and the integration might or might not recover at all.

Hope that this answer helps