Solved Examples: Laplace Transforms of Sine and Cosine in DE Applications

📚 Quick Study Guide

• 🔢 Definition: The Laplace transform of a function $f(t)$ is given by $F(s) = \mathcal{L}{f(t)} = \int_0^{\infty} e^{-st}f(t) dt$.
• 📈 Linearity: $\mathcal{L}{af(t) + bg(t)} = a\mathcal{L}{f(t)} + b\mathcal{L}{g(t)}$, where $a$ and $b$ are constants.
• 📐 Sine Function: $\mathcal{L}{\sin(at)} = \frac{a}{s^2 + a^2}$ for $s > 0$.
• 📉 Cosine Function: $\mathcal{L}{\cos(at)} = \frac{s}{s^2 + a^2}$ for $s > 0$.
• ⏱️ Time Invariance: $\mathcal{L}{f(t-a)u(t-a)} = e^{-as}F(s)$, where $u(t)$ is the unit step function.
• 💡 Differential Equations: Laplace transforms convert differential equations into algebraic equations, simplifying the solving process.
• 🧮 Inverse Laplace: After solving for $F(s)$, use inverse Laplace transform $\mathcal{L}^{-1}{F(s)}$ to obtain the solution $f(t)$.

Practice Quiz

1. What is the Laplace transform of $\sin(3t)$?
   1. $\frac{s}{s^2 + 9}$
   2. $\frac{3}{s^2 - 9}$
   3. $\frac{3}{s^2 + 9}$
   4. $\frac{s}{s^2 - 9}$
2. What is the Laplace transform of $\cos(2t)$?
   1. $\frac{2}{s^2 + 4}$
   2. $\frac{s}{s^2 - 4}$
   3. $\frac{s}{s^2 + 4}$
   4. $\frac{2}{s^2 - 4}$
3. Find the Laplace transform of $2\sin(t) + 3\cos(t)$.
   1. $\frac{2}{s^2 + 1} + \frac{3s}{s^2 + 1}$
   2. $\frac{2}{s^2 - 1} + \frac{3s}{s^2 - 1}$
   3. $\frac{3}{s^2 + 1} + \frac{2s}{s^2 + 1}$
   4. $\frac{3}{s^2 - 1} + \frac{2s}{s^2 - 1}$
4. What is $\mathcal{L}{\sin(at) + \cos(at)}$?
   1. $\frac{a + s}{s^2 + a^2}$
   2. $\frac{a - s}{s^2 + a^2}$
   3. $\frac{a + s}{s^2 - a^2}$
   4. $\frac{a - s}{s^2 - a^2}$
5. Determine the Laplace transform of $f(t) = 5\cos(5t)$.
   1. $\frac{5}{s^2 + 25}$
   2. $\frac{5s}{s^2 - 25}$
   3. $\frac{5s}{s^2 + 25}$
   4. $\frac{25}{s^2 + 25}$
6. Calculate $\mathcal{L}{3\sin(2t) - \cos(2t)}$.
   1. $\frac{6}{s^2 + 4} + \frac{s}{s^2 + 4}$
   2. $\frac{6}{s^2 - 4} - \frac{s}{s^2 - 4}$
   3. $\frac{6}{s^2 + 4} - \frac{s}{s^2 + 4}$
   4. $\frac{6}{s^2 - 4} + \frac{s}{s^2 - 4}$
7. What is the Laplace transform of $\cos(\omega t)$?
   1. $\frac{\omega}{s^2 + \omega^2}$
   2. $\frac{s}{s^2 - \omega^2}$
   3. $\frac{s}{s^2 + \omega^2}$
   4. $\frac{\omega}{s^2 - \omega^2}$