First-Order Equations with Exact Solutions

Most differential equations cannot be solved with a formula. A few important families can, and almost all of them are first order. This chapter is a toolbox for those families. Each tool rests on one idea: rewrite the equation so that one side becomes the derivative of something we already know, and then integrate.

You will meet the same equation in two costumes, the derivative form $y^{\prime }=f(x,y)$ and the differential form $M(x,y)dx+N(x,y)dy=0$. Choosing the right costume is often half of the work.

What you will learn

• 2.1 Exact equations. When $Mdx+Ndy$ is the total differential of a function $F(x,y)$, the solutions are simply the level curves $F(x,y)=c$. You will learn a one-line test for this and how to build $F$.
• 2.2 Separable and homogeneous equations. Put all the $y$'s on one side and all the $x$'s on the other, then integrate. A substitution $y=vx$ brings homogeneous equations into this form.
• 2.3 Linear and Bernoulli equations. A single multiplier, $e^{\int Pdx}$, solves every linear first-order equation. A power substitution turns Bernoulli equations into linear ones.
• 2.4 Special integrating factors and substitutions. How to repair an equation that is almost exact, and how a shift of the origin or a new variable turns a stubborn equation into one we already know.

Along the way we keep an eye on a quiet danger: dividing by an expression that can be zero. That step can make solutions disappear, and you will learn to find them again.

Prerequisites

You should be comfortable with integration techniques (substitution, integration by parts, partial fractions), with partial derivatives, and with the ideas of Chapter 1: what a solution is, what an initial-value problem is, and the existence and uniqueness theorem.

Sections

1. 2.1Exact Equations and Integrating Factors
2. 2.2Separable Equations and Equations Reducible to Separable Form
3. 2.3Linear Equations and Bernoulli Equations
4. 2.4Special Integrating Factors and Substitutions
5. QuizChapter 2 practice quiz: four new problems every time