Solved Examples of the Convolution Theorem in Differential Equations

📚 Quick Study Guide

• 🔍 The Convolution Theorem states that if $L{f(t)} = F(s)$ and $L{g(t)} = G(s)$, then $L{\int_{0}^{t} f(\tau)g(t-\tau) d\tau} = F(s)G(s)$.
• ⏱️ The convolution of two functions $f(t)$ and $g(t)$ is defined as $(f * g)(t) = \int_{0}^{t} f(\tau)g(t-\tau) d\tau$.
• 💡 Using Laplace transforms, differential equations can be transformed into algebraic equations, which are often easier to solve.
• 📝 The inverse Laplace transform of a product $F(s)G(s)$ can be found using the convolution theorem: $L^{-1}{F(s)G(s)} = \int_{0}^{t} f(\tau)g(t-\tau) d\tau$.
• ➗ When solving differential equations, the convolution theorem is particularly useful when dealing with non-homogeneous terms that are products of functions.
• 📈 Remember that $f * g = g * f$, meaning the order of convolution does not matter: $\int_{0}^{t} f(\tau)g(t-\tau) d\tau = \int_{0}^{t} g(\tau)f(t-\tau) d\tau$.

🧪 Practice Quiz

1. What is the convolution of $f(t) = t$ and $g(t) = 1$?
   1. $t^2$
   2. $\frac{t^2}{2}$
   3. $t$
   4. $1$
2. If $F(s) = \frac{1}{s}$ and $G(s) = \frac{1}{s+1}$, what is $L^{-1}{F(s)G(s)}$ using the convolution theorem?
   1. $1 - e^{-t}$
   2. $e^{-t}$
   3. $t e^{-t}$
   4. $t$
3. Compute the convolution of $f(t) = e^t$ and $g(t) = e^t$.
   1. $e^{2t}$
   2. $t e^t$
   3. $\frac{e^{2t}}{2}$
   4. $e^{2t} - 1$
4. What is the Laplace transform of $\int_{0}^{t} \sin(\tau) \cos(t-\tau) d\tau$?
   1. $\frac{1}{s^2 + 1}$
   2. $\frac{s}{s^2 + 1}$
   3. $\frac{s}{(s^2 + 1)^2}$
   4. $\frac{1}{(s^2 + 1)^2}$
5. Given $y'' + y = f(t)$, $y(0) = 0$, $y'(0) = 0$, express $y(t)$ using convolution.
   1. $\int_{0}^{t} f(\tau) \sin(t-\tau) d\tau$
   2. $\int_{0}^{t} f(\tau) \cos(t-\tau) d\tau$
   3. $\int_{0}^{t} f(\tau) e^{-(t-\tau)} d\tau$
   4. $\int_{0}^{t} f(\tau) d\tau$
6. Find $L^{-1}{\frac{1}{s^2(s^2+1)}}$ using convolution.
   1. $t - \sin(t)$
   2. $t - \cos(t)$
   3. $\sin(t) - t$
   4. $\cos(t) - t$
7. What is the convolution of $f(t) = t^2$ and $g(t) = 1$?
   1. $\frac{t^3}{3}$
   2. $t^3$
   3. $\frac{t^2}{2}$
   4. $2t$