Test questions on solving coupled linear ODEs using Laplace Transforms.

📚 Quick Study Guide

• 🔍 Laplace Transform Definition: The Laplace transform of a function $f(t)$ is given by $F(s) = \mathcal{L}{f(t)} = \int_0^{\infty} e^{-st}f(t) dt$.
• ⏱️ Laplace Transform of Derivatives: $\mathcal{L}{f'(t)} = sF(s) - f(0)$ and $\mathcal{L}{f''(t)} = s^2F(s) - sf(0) - f'(0)$.
• ➕ Linearity Property: $\mathcal{L}{af(t) + bg(t)} = a\mathcal{L}{f(t)} + b\mathcal{L}{g(t)}$, where $a$ and $b$ are constants.
• 🔄 Inverse Laplace Transform: The inverse Laplace transform, denoted by $\mathcal{L}^{-1}{F(s)}$, returns the original function $f(t)$ from its Laplace transform $F(s)$.
• 📝 Solving Coupled ODEs:
  1. Apply the Laplace transform to each equation in the system.
  2. Use initial conditions to simplify the transformed equations.
  3. Solve the resulting algebraic equations for the Laplace transforms of the unknown functions.
  4. Apply the inverse Laplace transform to find the solutions in the time domain.
• 💡 Common Laplace Transforms: $\mathcal{L}{1} = \frac{1}{s}$, $\mathcal{L}{e^{at}} = \frac{1}{s-a}$, $\mathcal{L}{\sin(at)} = \frac{a}{s^2 + a^2}$, $\mathcal{L}{\cos(at)} = \frac{s}{s^2 + a^2}$.

Practice Quiz

1. Question 1: What is the Laplace transform of $f(t) = t$?
   1. A) $\frac{1}{s}$
   2. B) $\frac{1}{s^2}$
   3. C) $\frac{1}{s+1}$
   4. D) $\frac{s}{s^2+1}$
2. Question 2: Given the system $\frac{dx}{dt} = -x + y$, $\frac{dy}{dt} = 2x - y$, with $x(0) = 1$ and $y(0) = 0$, what are the transformed equations for $X(s)$ and $Y(s)$ after applying the Laplace transform?
   1. A) $sX(s) - 1 = -X(s) + Y(s)$, $sY(s) = 2X(s) - Y(s)$
   2. B) $sX(s) = -X(s) + Y(s)$, $sY(s) = 2X(s) - Y(s)$
   3. C) $sX(s) - 1 = X(s) + Y(s)$, $sY(s) - 1 = 2X(s) - Y(s)$
   4. D) $sX(s) = X(s) + Y(s)$, $sY(s) = 2X(s) + Y(s)$
3. Question 3: What is the inverse Laplace transform of $F(s) = \frac{1}{s-3}$?
   1. A) $e^{-3t}$
   2. B) $e^{3t}$
   3. C) $\cos(3t)$
   4. D) $\sin(3t)$
4. Question 4: If $\mathcal{L}{x(t)} = X(s)$ and $\mathcal{L}{y(t)} = Y(s)$, and you have the equation $sX(s) + X(s) - Y(s) = 0$, what is the corresponding differential equation in the time domain?
   1. A) $\frac{dx}{dt} + x(t) - y(t) = 0$
   2. B) $\frac{dx}{dt} - x(t) + y(t) = 0$
   3. C) $x(t) + y(t) = 0$
   4. D) $\frac{dx}{dt} + y(t) = 0$
5. Question 5: Consider the coupled system: $\frac{dx}{dt} = x - 2y$, $\frac{dy}{dt} = 5x - y$. What are the eigenvalues of the matrix associated with this system?
   1. A) $2 \pm 3i$
   2. B) $1 \pm 2i$
   3. C) $0, 1$
   4. D) $2, -2$
6. Question 6: What is the Laplace transform of $\sin(at)$?
   1. A) $\frac{s}{s^2 + a^2}$
   2. B) $\frac{a}{s^2 + a^2}$
   3. C) $\frac{s}{s^2 - a^2}$
   4. D) $\frac{a}{s^2 - a^2}$
7. Question 7: Solve the following system using Laplace transform: $\frac{dx}{dt} + 2x = 0$, given $x(0) = 1$. What is $x(t)$?
   1. A) $e^{-2t}$
   2. B) $e^{2t}$
   3. C) $\cos(2t)$
   4. D) $\sin(2t)$