Chapter 9: Differential Equations

Students learn to identify the order and degree of a differential equation and explore solutions in both explicit and implicit forms. Direction fields are introduced as a graphical tool for visualizing solution curves without solving the equation analytically. The chapter covers separable differential equations, where variables can be isolated on opposite sides and integrated, with applications such as population growth, radioactive decay, and Newton’s Law of Cooling. First-order linear differential equations are solved using the integrating factor method, with examples drawn from mixing problems, exponential growth and decay, and cooling processes. The logistic differential equation is explored as a more realistic model for population growth with carrying capacity. Euler’s Method is introduced for numerical approximation of solutions when analytic methods are impractical. The chapter emphasizes interpreting solutions in context, validating models against real-world data, and understanding the limitations of different approaches. By the end, students can formulate, solve, and interpret first-order differential equations, setting the stage for higher-order and more complex systems in later studies.