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 $g(t)$ of large magnitude that acts over a short time interval about $t=c$ behave as impulse functions. Those are discontinue functions.

Definition: The unit impulse function $\delta$ is defined to have the properties $\delta (t)=0$ for $t\ne 0$, and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)dt=1$. In general, for a unit impulse at an arbitrary point $c$, $\delta (t-c)=0$ for $t\ne c$, and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t-c)dt=1$.

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

Example 1: Use Laplace Transform to solve the IVP. $y^{″}+4y^{\prime }+8y=\delta (t-2)$ $y(0)=1$, $y^{\prime }(0)=1$. 

Exercise 1: Use Laplace Transform to solve the IVP. $y^{″}+2y^{\prime }+5y=\delta (t-1)$, $y(0)=1$, $y^{\prime }(0)=2$. 

Example 2: Use Laplace Transform to solve the IVP. $y^{″}+9y=\delta (t-4)-\delta (t-8)$, $y(0)=0$, $y^{\prime }(0)=1$. 

Exercise 2: Use Laplace Transform to solve the IVP. $y^{″}+4y=-\delta (t-3)+\delta (t-2)$, $y(0)=1$, $y^{\prime }(0)=0$. 

Example 3: Use Laplace Transform to solve the IVP. $y^{″}-y^{\prime }-2y=\delta (t-3)+u_4(t)$, $y(0)=0$, $y^{\prime }(0)=1$. 

Exercise 3: Use Laplace Transform to solve the IVP. $y^{″}+2y^{\prime }-8y=\delta (t-5)-u_3(t)$, $y(0)=1$, $y^{\prime }(0)=1$. 

Example 4: Use Laplace Transform to solve the IVP. $y^{″}+2y^{\prime }+3y=\delta (t-3\pi )+\text{sin}(t)$, $y(0)=0$, $y^{\prime }(0)=0$. 

Exercise 4: Use Laplace Transform to solve the IVP. $y^{″}+4y^{\prime }+13y=\delta (t-\pi )-\text{cos}(t)$, $y(0)=0$, $y^{\prime }(0)=0$. 

Group Work

1. Use Laplace Transform to solve the IVP. $y^{″}+2y^{\prime }+5y=u_2(t)-3\delta (t-4)$, $y(0)=6$, $y^{\prime }(0)=0$. 

2. Use Laplace Transform to solve the IVP. $y^{″}+4y^{\prime }+4y=e^{2t}-\delta (t-e)$, $y(0)=0$, $y^{\prime }(0)=1$.