Chapter Outline

This chapter delves into first-order differential equations, vital in science and engineering for modeling rates of change in numerous phenomena. It covers their structure, solution techniques, and real-world applications in fields like population dynamics, thermal processes, and electrical circuits.

2.1 Separable First-Order Differential Equations: This section addresses separable differential equations, a category of first-order equations where each variable can be separated on different sides of the equation.

2.2 Linear First-Order Differential Equations: This section covers the solution to first-order nonhomogeneous linear equations.

2.3 Exact Differential Equations: This part explains the criteria for an equation to be exact and outlines methods for solving these equations.

2.4 Integrating Factors: This section explores the techniques of utilizing integrating factors to transform a non-exact equation into an exact equation that can be solved.

2.5 Applications of First-Order Differential Equations: The final section illustrates the use of first-order differential equations in modeling growth and decay, substance mixing, temperature changes, motion under gravity, and circuit behaviors.

Pioneers of Progress

Mary Cartwright, born in 1900 in Aynho, Northamptonshire, England, emerged as a pioneering mathematician in an era when female academics were a rarity. Her journey in mathematics began at Oxford University, leading her to Cambridge, where she initially focused on classical analysis. However, it was during World War II, while investigating the problem of radio waves and their interference patterns, that Cartwright made a groundbreaking discovery. Collaborating with J.E. Littlewood, she delved into nonlinear differential equations, and their work laid the foundational stones for what would later be known as chaos theory. Cartwright’s foray into this field produced seminal results, including the Cartwright-Littlewood theorem and her study of the Van der Pol oscillator, a concept critical in the understanding of oscillatory systems. Her extraordinary contributions not only advanced the field of mathematics but also broke gender barriers, setting a precedent for women in STEM. Mary Cartwright’s life was a blend of intellectual rigor and quiet resilience, inspiring a legacy that continues to encourage mathematicians, especially women, to explore and reshape the boundaries of mathematical knowledge.