Chapter Outline

This chapter provides an overview of fundamental concepts in differential equations along with an introduction to direction fields for first-order differential equations.

1.1 Introduction: This section covers basic definitions concerning differential equations, including their order, various classifications, and the nature of their solutions.

1.2 Direction Fields: This section briefly introduces direction fields, a tool for visually representing the behavior of solutions to first-order differential equations without needing an exact solution formula.

Pioneers of Progress

Émilie du Châtelet, born in Paris in 1706, was a woman of exceptional intellect and determination who carved her unique path in the male-dominated world of science and mathematics during the Enlightenment. Despite societal norms restricting women’s access to formal education, Du Châtelet educated herself in mathematics and physics, often through creative means such as disguising herself as a man to attend lectures. Her most significant work, a translation and commentary on Isaac Newton’s ‘Principia Mathematica’, remains the standard French translation to this day. In it, she clarified Newton’s ideas and expanded on them, particularly in her elucidation of the principle of conservation of energy. Émilie du Châtelet’s work laid the groundwork for future developments in physics and mathematics, including those in differential equations. Her tenacity and brilliance broke through the constraints of her time, paving the way for future generations of women in science, and her legacy continues to inspire and challenge norms in the scientific community.