University differential equations exam prep: Impulsive forcing IVPs

📚 Quick Study Guide

• ⏱️ Impulsive forcing involves functions like the Dirac delta function, $\delta(t-a)$, representing an impulse at $t=a$.
• ➗ When solving IVPs with impulsive forcing, integrate the differential equation over a small interval around the impulse to find the jump in the dependent variable's derivative.
• 📝 Remember that $\int_{-\epsilon}^{\epsilon} \delta(t) f(t) dt = f(0)$, which is essential for evaluating the effect of the impulse.
• 💡 The Laplace transform of the Dirac delta function is $L{\delta(t-a)} = e^{-as}$.
• 📈 After applying the Laplace transform, solve for $Y(s)$ and use inverse Laplace transforms to find the solution $y(t)$.
• 📌 Always consider the initial conditions and how they are affected by the impulsive force.
• 🧮 Piecewise functions often arise when dealing with impulsive forcing, so make sure to define the solution for different time intervals.

Practice Quiz

1. What is the Laplace transform of the Dirac delta function $\delta(t-3)$?
   1. $e^{-3s}$
   2. $1$
   3. $s e^{-3s}$
   4. $\frac{1}{s-3}$
2. Consider the IVP $y'' + 4y = \delta(t-\pi)$, $y(0) = 0$, $y'(0) = 0$. What is $Y(s)$, the Laplace transform of $y(t)$?
   1. $\frac{e^{-\pi s}}{s^2 + 4}$
   2. $\frac{1}{s^2 + 4}$
   3. $\frac{e^{-\pi s}}{s+4}$
   4. $\frac{s e^{-\pi s}}{s^2 + 4}$
3. What is the value of the integral $\int_{-1}^{1} \delta(t) t^2 dt$?
   1. $0$
   2. $1$
   3. $-1$
   4. $2$
4. Solve for $y(t)$ given $Y(s) = \frac{e^{-2s}}{s+1}$.
   1. $u_2(t) e^{-(t-2)}$
   2. $e^{-t}$
   3. $u_2(t) e^{-t}$
   4. $e^{-(t+2)}$
5. Which of the following differential equations represents an impulsive force applied at $t=5$ with magnitude 2?
   1. $y'' + y = \delta(t-5)$
   2. $y'' + y = 2\delta(t)$
   3. $y'' + y = 2\delta(t-5)$
   4. $y'' + y = \delta(t)$
6. What is the solution to $y' = \delta(t)$, $y(0) = 0$?
   1. $u_0(t)$
   2. $1$
   3. $0$
   4. $t$
7. If $y'' + 9y = \delta(t- \frac{\pi}{2})$ and $y(0)=0$, $y'(0)=0$, find $y(t)$ for $t > \frac{\pi}{2}$.
   1. $\frac{1}{3} \sin(3t)$
   2. $\frac{1}{3} \sin(3(t - \frac{\pi}{2}))$
   3. $ \sin(3(t - \frac{\pi}{2}))$
   4. $\frac{1}{3} \cos(3t)$