Section 4.5 Impulse functions

Objective:

1. Definition of the impulse function

2. Use Laplace transformation to solve differential equations with impulse function

We are introducing a new function, impulse functions. An electrical circuit or mechanical system subject to a sudden voltage or force [latex]g(t)[/latex] of large magnitude that acts over a short time interval about [latex]t=c[/latex] behave as impulse functions. Those are discontinue functions.

Definition: The unit impulse function [latex]\delta[/latex] is defined to have the properties [latex]\delta(t)=0[/latex] for [latex]t\neq0[/latex], and [latex]\int_{-\infty}^{\infty}\delta(t)dt=1[/latex]. In general, for a unit impulse at an arbitrary point [latex]c[/latex], [latex]\delta(t-c)=0[/latex] for [latex]t\neq c[/latex], and [latex]\int_{-\infty}^{\infty}\delta(t-c)dt=1[/latex].

Theorem: Laplace Transform of [latex]\delta(t-c)[/latex] is [latex]e^{-cs}[/latex]. The proof for [latex]\delta(t)[/latex], i.e. [latex]c=0[/latex] is at the page 32, 33, 34, and 35 here. 

Example 1: Use Laplace Transform to solve the IVP. [latex]y''+4y'+8y=\delta(t-2)[/latex] [latex]y(0)=1[/latex], [latex]y'(0)=1[/latex]. 

Exercise 1: Use Laplace Transform to solve the IVP. [latex]y''+2y'+5y=\delta(t-1)[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=2[/latex]. 

Example 2: Use Laplace Transform to solve the IVP. [latex]y''+9y=\delta(t-4)-\delta(t-8)[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=1[/latex]. 

Exercise 2: Use Laplace Transform to solve the IVP. [latex]y''+4y=-\delta(t-3)+\delta(t-2)[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=0[/latex]. 

Example 3: Use Laplace Transform to solve the IVP. [latex]y''-y'-2y=\delta(t-3)+u_{4}(t)[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=1[/latex]. 

Exercise 3: Use Laplace Transform to solve the IVP. [latex]y''+2y'-8y=\delta(t-5)-u_{3}(t)[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=1[/latex]. 

Example 4: Use Laplace Transform to solve the IVP. [latex]y''+2y'+3y=\delta(t-3\pi)+\text{sin}(t)[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=0[/latex]. 

Exercise 4: Use Laplace Transform to solve the IVP. [latex]y''+4y'+13y=\delta(t-\pi)-\text{cos}(t)[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=0[/latex]. 

Group Work

1. Use Laplace Transform to solve the IVP. [latex]y''+2y'+5y=u_{2}(t)-3\delta(t-4)[/latex], [latex]y(0)=6[/latex], [latex]y'(0)=0[/latex]. 

2. Use Laplace Transform to solve the IVP. [latex]y''+4y'+4y=e^{2t}-\delta(t-e)[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=1[/latex].