Applications of First-Order Equations
Chapter 2 gave us methods for solving first-order equations. This chapter puts them to work. Each section starts from a physical or geometric question, turns it into a differential equation, solves it, and then reads the answer back in the language of the problem: metres per second, degrees Celsius, kilograms of salt, years.
The mathematics is mostly familiar. Almost every equation here is separable or linear. The new skill is modelling: choosing the unknown and its units, writing down the right rates or forces with the right signs, and using measured data to find the constants. We also learn to check answers the way engineers do: do the units match, and does the solution behave sensibly for large times?
What you will learn
- 3.1 Orthogonal and oblique trajectories. How a family of curves defines a differential equation, and how to find the curves that cross the family at right angles (like field lines and equipotentials) or at any fixed angle. Includes the polar-coordinate version.
- 3.2 Problems in mechanics. Newton's second law as a first-order equation for velocity: free fall, linear and quadratic air resistance, terminal velocity, throwing upward, and sliding down an inclined plane with friction.
- 3.3 Rate problems. Exponential growth and decay with half-life, Newton's law of cooling, salt in a stirred tank with equal or unequal flow rates, logistic population growth, and simple chemical reaction rates.
Each section has interactive simulators. Change the data and watch how the solution responds; this is the quickest way to build intuition for time constants and equilibrium values.
Prerequisites
You should be able to solve separable, homogeneous, exact and linear first-order equations (Sections 2.1–2.3), and be comfortable with partial fractions, exponentials and logarithms. For Section 3.2, basic mechanics (forces, Newton's second law, resolving a force into components) is helpful.