Solved examples of Existence and Uniqueness Theorem for Linear IVPs

📚 Quick Study Guide

• 🔑 Linear First-Order IVP Form: A first-order linear initial value problem (IVP) has the form $y' + p(t)y = g(t)$, with initial condition $y(t_0) = y_0$.
• ✅ Existence Condition: If $p(t)$ and $g(t)$ are continuous on an open interval $I$ containing $t_0$, then a solution exists on that interval.
• ✨ Uniqueness Condition: If $p(t)$ and $g(t)$ are continuous on an open interval $I$ containing $t_0$, the solution is unique on that interval.
• 🧭 Theorem's Core Idea: The theorem guarantees a solution exists and it's the only one, *provided* the coefficients in the differential equation are nice (continuous) near the initial point.
• 📏 Interval of Existence/Uniqueness: The largest interval around $t_0$ where both $p(t)$ and $g(t)$ are continuous.

✏️ Practice Quiz

1. Question 1: Consider the IVP $ty' + 2y = 4t^2$, $y(1) = 2$. What is the largest interval where the Existence and Uniqueness Theorem guarantees a unique solution?
   1. $t > 0$
   2. $-\infty < t < \infty$
   3. $t < 0$
   4. $t \neq 0$
2. Question 2: For the IVP $(t-3)y' + y = t^2$, $y(1) = 5$, what is the largest interval for which a unique solution is guaranteed?
   1. $-\infty < t < 3$
   2. $3 < t < \infty$
   3. $-\infty < t < \infty$
   4. $1 < t < 3$
3. Question 3: Consider $y' + \frac{1}{t}y = \frac{1}{t^2 - 4}$, $y(1) = 0$. What is the interval of existence and uniqueness?
   1. $t > 2$
   2. $-2 < t < 2$
   3. $0 < t < 2$
   4. $t < -2$
4. Question 4: For $y' + y \tan(t) = \sin(t)$, $y(0) = 1$, what is the interval of existence and uniqueness?
   1. $-\frac{\pi}{2} < t < \frac{\pi}{2}$
   2. $0 < t < \frac{\pi}{2}$
   3. $-\infty < t < \infty$
   4. $0 < t < \pi$
5. Question 5: Which condition is essential for the Existence and Uniqueness Theorem to apply to a linear IVP?
   1. The initial condition must be zero.
   2. The functions $p(t)$ and $g(t)$ must be continuous.
   3. The differential equation must be homogeneous.
   4. The differential equation must have constant coefficients.
6. Question 6: Consider the IVP $y' + \frac{1}{t-2}y = e^t$, $y(0) = 1$. What is the interval where a unique solution exists?
   1. $-\infty < t < 2$
   2. $2 < t < \infty$
   3. $-\infty < t < \infty$
   4. $t > 0$
7. Question 7: What does the Existence and Uniqueness Theorem guarantee?
   1. That a solution exists.
   2. That a solution is unique.
   3. That a solution exists and is unique.
   4. Nothing about the solution.