Dive into a comprehensive video lecture on differential equations, focusing on the Existence and Uniqueness Theorem from Nagle's Section 1.2. Learn to verify solutions for various differential equations, including second-order and complex cases. Explore scenarios where real-valued solutions do not exist. Master the application of the Existence and Uniqueness Theorem through multiple examples, culminating in verifying a solution to the differential equation y''+y=tan(x). Enhance your understanding of these fundamental concepts in calculus and differential equations over the course of 90 minutes.