Differential Equations and Their Solutions
Almost every law of engineering and physics describes how something changes: how fast a capacitor charges, how quickly a hot part cools, how a spring pushes back when it is stretched. Written in mathematical form, such laws become differential equations, equations whose unknown is a function and which involve its derivatives. This first chapter builds the vocabulary and the basic ideas that the rest of the course relies on.
What you will learn
- Section 1.1 shows how to classify an equation: ordinary or partial, its order, and whether it is linear or nonlinear. It also shows where differential equations come from, both from physical laws (cooling, circuits, springs, draining tanks) and from families of curves.
- Section 1.2 explains what it means to solve a differential equation. You will meet explicit, implicit and "formal" solutions, see why every solution lives on an interval, and learn how an equation of order usually has an -parameter family of solutions. Direction fields give a picture of all solutions at once.
- Section 1.3 adds conditions to pick out a single solution: initial-value problems and boundary-value problems. The central result is the existence and uniqueness theorem, which tells you when a problem is guaranteed to have exactly one solution, and examples show what can go wrong when its hypotheses fail.
Interactive tools let you draw direction fields, check candidate solutions step by step, and test the hypotheses of the existence and uniqueness theorem on your own equations.
Prerequisites
You need single-variable calculus: derivatives (product, quotient and chain rules), implicit differentiation, basic integration and the fundamental theorem of calculus. You should also be comfortable with partial derivatives and continuity of functions of two variables, which appear in Section 1.3. No earlier knowledge of differential equations is assumed.