Introduction to Differential Equations: Foundations and First-Order Equations

Differential equations are the mathematical language used to describe how the world changes, from physics to biology and finance. If you are preparing for high school exams or introductory college calculus, understanding how to set up and solve these equations is essential. This text-based course guides you through the fundamental principles of ordinary differential equations, focusing on first-order equations. You will transition from understanding basic rates of change to confidently solving mathematical models of real-world phenomena. What you'll learn: - Understand key terminology including order, degree, and the difference between general and particular solutions. - Solve first-order differential equations using the variable separable method. - Apply integrating factors to solve linear first-order differential equations. - Formulate mathematical models for population growth, decay, and heating processes. - Practice solving homogeneous differential equations step-by-step. - Explore basic numerical approximation techniques like Euler's method for real-world scenarios. The course begins with foundational definitions and classification of equations before moving systematically through analytical solution methods and practical modeling applications. Designed specifically for beginners with a basic background in single-variable calculus, this course requires no prior exposure to differential equations. Start building your mathematical foundation and unlock the tools to analyze dynamic systems.