Pleshkov Denis/preprint: Added The Impulse–Initial Condition Equivalence in Linear ODEs: A Literature Review

Abstract This literature review examines how linear time-invariant (LTI) ordinary differential equations (ODEs) with impulsive forcing (represented by the Dirac delta function and its derivatives) are treated across differential equations, vibration theory, and control theory. Motivated by the practical need to find proper solution approaches and closed-form formulas for such systems, we surveyed 100+ sources and classified them by how explicitly each addresses the equivalence between an impulse-forced initial value problem and a homogeneous one with modified initial conditions. While this equivalence is well established for specific low-order equations, no source in the literature provides a general closed-form solution for an LTI ODE forced by a sum of derivatives of the Dirac delta — the central gap this review identifies, alongside the broader lack of a unified computational framework spanning all derivative and equation orders. Keywords Literature review, delta function, differential equation, ODE, impulse response, initial condition, equivalence