Test Questions on Laplace Transform Method for Solving ODEs

📚 Quick Study Guide

• ➗ The Laplace Transform converts a differential equation into an algebraic equation, which is often easier to solve.
• ⏱️ The Laplace Transform of a function $f(t)$ is defined as $F(s) = \int_0^{\infty} e^{-st}f(t) dt$.
• 🔑 Key Laplace Transforms to remember:
  • ⚡ $L{1} = \frac{1}{s}$
  • 📈 $L{t} = \frac{1}{s^2}$
  • 💡 $L{e^{at}} = \frac{1}{s-a}$
  • 🧮 $L{\sin(at)} = \frac{a}{s^2 + a^2}$
  • 🧪 $L{\cos(at)} = \frac{s}{s^2 + a^2}$
• 📝 Important Properties:
  • 📏 Linearity: $L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}$
  • 🌱 First Shifting Theorem: $L{e^{at}f(t)} = F(s-a)$
  • 🍁 Transform of Derivatives: $L{f'(t)} = sF(s) - f(0)$, $L{f''(t)} = s^2F(s) - sf(0) - f'(0)$
• ↔️ Inverse Laplace Transform: Finding $f(t)$ from $F(s)$, denoted as $L^{-1}{F(s)} = f(t)$. Partial fraction decomposition is often used.
• 💡 Solving ODEs using Laplace Transforms involves transforming the ODE, solving for $Y(s)$ (where $Y(s)$ is the Laplace Transform of $y(t)$), and then finding the inverse Laplace Transform to get $y(t)$.

Practice Quiz

1. What is the Laplace Transform of $f(t) = t^2$?
   1. $\frac{1}{s^3}$
   2. $\frac{2}{s^3}$
   3. $\frac{6}{s^3}$
   4. $\frac{1}{s^2}$
2. What is the Laplace Transform of $f(t) = e^{3t}$?
   1. $\frac{1}{s+3}$
   2. $\frac{1}{s-3}$
   3. $\frac{s}{s-3}$
   4. $\frac{s}{s+3}$
3. What is the Laplace Transform of $f(t) = \sin(2t)$?
   1. $\frac{s}{s^2 + 4}$
   2. $\frac{2}{s^2 - 4}$
   3. $\frac{s}{s^2 - 4}$
   4. $\frac{2}{s^2 + 4}$
4. What is the inverse Laplace Transform of $F(s) = \frac{1}{s-5}$?
   1. $e^{-5t}$
   2. $\cos(5t)$
   3. $e^{5t}$
   4. $\sin(5t)$
5. What is the Laplace Transform of $f'(t)$ if $f(0) = 2$ and $F(s)$ is the Laplace transform of $f(t)$?
   1. $sF(s) + 2$
   2. $F(s) - 2$
   3. $sF(s) - 2$
   4. $2F(s) - s$
6. What is the Laplace Transform of $f(t) = 5t + 3$?
   1. $\frac{5}{s^2} + \frac{3}{s}$
   2. $\frac{5}{s} + \frac{3}{s^2}$
   3. $\frac{5}{s^2} + \frac{3}{s^2}$
   4. $\frac{8}{s^2}$
7. Solve for $Y(s)$, the Laplace transform of $y(t)$, given the differential equation $y' + 2y = 0$ and $y(0) = 1$.
   1. $Y(s) = \frac{1}{s+2}$
   2. $Y(s) = \frac{s}{s+2}$
   3. $Y(s) = \frac{1}{s-2}$
   4. $Y(s) = \frac{s}{s-2}$