∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 4 Theory

Laplace Transforms & Step/Impulse Forcing Functions

Converting differential equations in time t into algebraic equations in s

Visualize how a Heaviside step forcing function u_c(t) switches on at t = c and triggers a transient oscillatory response.

1. Why the Laplace Transform Turns Differentiation into Multiplication

Integrating by parts shows that L{f'(t)} = sF(s) - f(0). Discontinuous step inputs u_c(t) and Dirac delta hammer blows δ(t - c) become simple exponential factors e^(-cs) in the s-domain.

L{y′′}=s2Y(s)−sy(0)−y′(0),L{δ(t−c)}=e−cs\mathcal{L}\{y''\} = s^2 Y(s) - sy(0) - y'(0), \qquad \mathcal{L}\{\delta(t - c)\} = e^{-cs}

Quick Reference & Cheat Sheet

Fundamental Laplace Transform Pairs
L{eat}=1s−a,L{sin⁡(ωt)}=ωs2+ω2,L{cos⁡(ωt)}=ss2+ω2\mathcal{L}\{e^{at}\} = \frac{1}{s-a}, \quad \mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2+\omega^2}, \quad \mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2+\omega^2}
Maps time-domain exponentials and sinusoids to rational poles in the complex s-plane.