Introduction to Differential Equations

Mathematical modeling is an attempt to describe how the real world works using mathematics and symbols. Models help us understand and predict the behavior of a system, or investigate how various components affect that behavior. Because of this, mathematical modeling is used throughout the natural sciences (physics, chemistry, biology), engineering, the social sciences (sociology, economics, psychology), and even music and linguistics.

In many situations, a mathematical model involves the search for an unknown function whose derivatives satisfy an equation. Such equations are called differential equations. We naturally want to find a solution of a given differential equation. However, you will discover as the course progresses that studying differential equations entails more than merely knowing techniques developed by others to solve them.

Since learning the terminology of any specialized subject is essential for studying it, we start with definitions and some basic concepts.

What This Chapter Covers

Section	What you will learn
1.1 What Is a Differential Equation?	The definition of a differential equation, the difference between ordinary and partial differential equations, and prime and dot notation for derivatives.
1.2 Classification of Differential Equations	Order, degree, linear versus nonlinear, homogeneous versus non-homogeneous, and explicit versus implicit solutions.
1.3 Family of Curves	How a family of curves containing arbitrary constants corresponds to a differential equation, and how to eliminate the constants to find it.
1.4 Initial Value Problems vs. Boundary Value Problems	How side conditions imposed at one point differ from conditions imposed at two endpoints, and why the distinction changes the solution methods.

Why Differential Equations Matter

Almost every quantitative law of nature is stated as a differential equation, because laws typically describe rates of change rather than values.

• Mechanics. Newton's second law $F=ma$ is a second-order differential equation for position, since acceleration is the second derivative of position with respect to time.
• Population biology. The logistic equation ${\displaystyle \frac{dP}{dt}}=rP(1-{\displaystyle \frac{P}{K}})$ models a population that grows quickly when small and levels off near a carrying capacity $K$.
• Electrical circuits. A series $RLC$ circuit satisfies $L{\displaystyle \frac{d^{2}q}{d{t}^2}}+R{\displaystyle \frac{dq}{dt}}+{\displaystyle \frac{q}{C}}=E(t)$, the same equation form as a damped, driven mechanical oscillator.
• Heat and diffusion. The heat equation ${\displaystyle \frac{\partial u}{\partial t}}=\rho {\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}$ describes how temperature spreads through a body over time.
• Finance. Continuously compounded interest satisfies ${\displaystyle \frac{dA}{dt}}=rA$, whose solution $A=A_0{e}^{rt}$ is the exponential growth law.

Frequently Asked Questions

What is the difference between a differential equation and an ordinary algebraic equation?

In an algebraic equation such as $x^2-5x+6=0$ the unknown is a number, and a solution is a specific value like $x=2$. In a differential equation the unknown is a function, and a solution is an entire function such as $y=\sin x$. Because a differential equation constrains the function through its derivatives, its solutions usually form a whole family rather than a short list of numbers.

Do I need to know how to solve every differential equation?

No, and in fact most differential equations cannot be solved in closed form using elementary functions. A large part of the subject consists of recognizing which special forms can be solved, and of extracting qualitative information (does a solution exist, is it unique, is it stable, what does it do for large $t$) when a formula is out of reach.

How many arbitrary constants does the general solution have?

An ordinary differential equation of order $n$ has, in general, a solution containing $n$ arbitrary constants. A first-order equation carries one constant, a second-order equation carries two, and so on. Each side condition you impose, such as an initial value, pins down one of those constants.

What background do I need for this book?

Single-variable differential and integral calculus is essential: derivatives, the chain rule, implicit differentiation, techniques of integration, and partial fractions. Later chapters use complex numbers, linear algebra (matrices, determinants, eigenvalues, and eigenvectors), and power series. The Prerequisite Math chapter reviews complex numbers if you need a refresher.

Is an equation with partial derivatives covered here?

Only in passing. This book is about ordinary differential equations, where the unknown function depends on a single independent variable. Partial differential equations appear in Section 1.1 and Section 1.2 so that you can tell the two kinds apart, and the boundary value problem chapter points toward the settings where they arise, but their solution methods belong to a separate course.