13: Qualitative Methods for Differential Equations

Last updated
Save as PDF

Page ID: 121151

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Not all differential equations are easily solved analytically. Furthermore, even when we find the analytic solution, it is not necessarily easy to interpret, graph, or understand. This situation motivates qualitative methods that promote an overall understanding of behavior - directly from information in the differential equation - without the challenge of finding a full functional form of the solution.

In this chapter we expand our familiarity with differential equations and assemble new, qualitative techniques for understanding them. We consider differential equations in which the expression on one side, \(f(y)\), is nonlinear, i.e. equations of the form

\[\frac{d y}{d t}=f(y) \nonumber \]

in which \(f\) is more complicated than the form \(a-b y\). Geometric techniques, rather than algebraic calculations form the core of the concepts we discuss.

13.1: Linear and Nonlinear Differential Equations
13.2: The Geometry of Change
In this section, we introduce a new method for understanding differential equations using graphical and geometric arguments. Such methods circumvent the solutions that we expressed in terms of analytic formulae. We resort to concepts learned much earlier - for example, the derivative as a slope of a tangent line - in order to use the differential equation itself to assemble a sketch of the behavior that it predicts.
13.3: Applying qualitative analysis to biological models
The qualitative ideas developed so far will now be applied to to problems from biology. In the following sections we first use these methods to obtain a thorough understanding of logistic population growth. We then derive a model for the spread of a disease, and use qualitative arguments to analyze the predictions of that differential equation model.
13.4: Summary
13.5: Exercises