Test Questions: Applying the First Shifting Theorem for Inverse Laplace Transforms in DE

📚 Quick Study Guide

• 🔑 The First Shifting Theorem states that if $\mathcal{L}^{-1}{F(s)} = f(t)$, then $\mathcal{L}^{-1}{F(s-a)} = e^{at}f(t)$.
• 🔢 To apply the theorem, manipulate the given expression to match the form $F(s-a)$. Identify $F(s)$ and $a$.
• 🔄 Find the inverse Laplace transform of $F(s)$, which is $f(t)$.
• 🚀 Multiply $f(t)$ by $e^{at}$ to obtain the final inverse Laplace transform: $e^{at}f(t)$.
• 📝 Remember to complete the square in the denominator when necessary to match the required form.

Practice Quiz

1. Question 1: What is the inverse Laplace transform of $\frac{1}{s-3}$?
1. $e^{3t}$
2. $e^{-3t}$
3. $t e^{3t}$
4. $t e^{-3t}$
2. Question 2: Find the inverse Laplace transform of $\frac{2}{(s+2)^2}$.
1. $2te^{-2t}$
2. $2te^{2t}$
3. $e^{-2t}$
4. $e^{2t}$
3. Question 3: What is the inverse Laplace transform of $\frac{1}{s^2 + 2s + 2}$?
1. $e^{-t}\sin(t)$
2. $e^{t}\sin(t)$
3. $e^{-t}\cos(t)$
4. $e^{t}\cos(t)$
4. Question 4: Determine the inverse Laplace transform of $\frac{s}{s^2 + 4s + 5}$.
1. $e^{-2t}(\cos(t) - 2\sin(t))$
2. $e^{2t}(\cos(t) - 2\sin(t))$
3. $e^{-2t}(\cos(t) + 2\sin(t))$
4. $e^{2t}(\cos(t) + 2\sin(t))$
5. Question 5: Calculate the inverse Laplace transform of $\frac{1}{(s-1)^3}$.
1. $\frac{t^2e^{t}}{2}$
2. $\frac{t^2e^{-t}}{2}$
3. $t^2e^{t}$
4. $t^2e^{-t}$
6. Question 6: What is the inverse Laplace transform of $\frac{s+2}{s^2 + 2s + 5}$?
1. $e^{-t}\cos(2t)$
2. $e^{t}\cos(2t)$
3. $e^{-t}\sin(2t)$
4. $e^{t}\sin(2t)$
7. Question 7: Find the inverse Laplace transform of $\frac{1}{s^2 + 6s + 13}$.
1. $\frac{1}{2}e^{-3t}\sin(2t)$
2. $\frac{1}{2}e^{3t}\sin(2t)$
3. $e^{-3t}\sin(2t)$
4. $e^{3t}\sin(2t)$