8: First Order Equations — Read

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In this chapter, we study first-order differential equations for which there are general methods of solution.

8.1: Linear First Order Equations
This section deals with linear equations, the simplest kind of ﬁrst order equations. In this section we introduce the method of variation of parameters. The idea underlying this method will be a unifying theme for our approach to solving many different kinds of differential equations throughout the book.
- 8.1.1: Linear First Order Equations (Exercises)
8.2: Separable Equations
The section deals with separable equations, the simplest nonlinear equations. In this section we introduce the idea of implicit and constant solutions of differential equations, and we point out some differences between the properties of linear and nonlinear equations.
- 8.2.1: Separable Equations (Exercises)
8.3: Existence and Uniqueness of Solutions of Nonlinear Equations
Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for nonlinear equations. In this section we state such a condition and illustrate it with examples.
- 8.3.1: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)
8.4: Transformation of Nonlinear Equations into Separable Equations
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
- 8.4.1: Transformation of Nonlinear Equations into Separable Equations (Exercises)
8.5: Exact Equations
This section covers exact differential equations, which are given this name because the method for solving them uses the idea of an exact differential from calculus.
- 8.5.1: Exact Equations (Exercises)
8.6: Integrating Factors
This section deals with equations that are not exact, but can made exact by multiplying them by a function known called integrating factor.
- 8.6.1: Integrating Factors (Exercises)

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