Differential Equations and The - Braun, Martin 7579 · Problem 2.12.Ex3

Introduction to Applied Nonlinear Dynamical Systems and Chaos — Braun, Martin 7579

Mathematics — Calculus · dirac-delta-impulse

Problem

(a) Solve the initial-value problem

(d^2y/dt^2)-3(dy/dt)+2y=f(t)

;

y(0)=1

y'(0)=0

, where

f(t)

is an impulsive function which acts on the extremely short time interval

2\leq t\leq 2+\tau

, and

\int_2^{2+\tau}f(t)\,dt=-1

. (b) Solve the initial-value problem

(d^2y/dt^2)-3(dy/dt)+2y=0

;

y(0)=1

y'(0)=0

, on the interval

0\leq t\leq 2

. Compute

z_0=y(2)

and

z_0'=y'(2)

. Then solve the initial-value problem

\frac{d^2y}{dt^2}-3\frac{dy}{dt}+2y=0; \quad y(2)=z_0,\; y'(2)=z_0'-1, \quad 2<t<\infty.

Compare this solution with the solution of part (a).

Intuitive explanation

The key insight is that an impulsive force acting over an infinitesimally short interval is modeled by the Dirac delta function, which creates a jump discontinuity in velocity at the moment of impact, while position remains continuous. In part (b), you'll solve the homogeneous equation in two phases—before and after the impulse at $t=2$ —and manually apply the velocity jump $\Delta y' = -1$ as a boundary condition. This piecewise approach should yield the same result as part (a), illustrating how the delta function encodes an instantaneous momentum change into the differential equation.

Intuitive explanation

Related theory

The Dirac Delta Function And Distributions

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