Chapter 8

Approximate Methods for First-Order Equations

The methods of Chapters 2 and 4 work only for special equations. Most equations met in practice, such as $y^{'} = x^{2} + sin y$ , have no solution formula at all. This chapter shows how to learn about their solutions anyway, and how to compute them to any accuracy we need. This is how engineering software actually solves differential equations.

The chapter moves from qualitative to quantitative tools:

8.1 Graphical methods: line elements, direction fields and isoclines; locating extrema and inflection points without solving.
8.2 Power series methods: Taylor polynomials of the solution, by repeated differentiation or by undetermined coefficients.
8.3 Successive approximations: Picard's iteration, and why the Lipschitz condition guarantees existence and uniqueness.
8.4 Euler's method: the simplest step-by-step numerical method, and the different kinds of error: local, global and round-off.

Sections 8.5 to 8.7 develop more accurate numerical methods: improved Euler, Runge–Kutta and Adams–Bashforth/Adams–Moulton. Section 8.8 extends them to higher-order equations and systems.

What you will be able to do

Sketch solution curves of $y^{'} = f (x, y)$ from its isoclines, and predict their long-term behaviour.
Compute series and Picard approximations near an initial point, and judge where they can be trusted.
Carry out a numerical method by hand or with a calculator, estimate its error, and choose a sensible step size.

Prerequisites

Chapter 1 (solutions, direction fields, the existence–uniqueness theorem) and the first-order methods of Chapter 2. From calculus: the chain rule, partial derivatives, Taylor's formula with remainder, and power series. Section 8.8 also uses the idea of a system of first-order equations (Chapter 7).