CHAPTER 6
Differential Equations
6.1 A Modeling Introduction to Differential Equations
6.2 Solutions and Separable Equations
6.4 Slope Fields and Euler's Method
Figure 6.1 We are able to think, move, and eat because of collections of neurons in our bodies. This photograph shows the neurons of a ground squirrel.
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“Among all the mathematical disciplines the theory of differential equations is the most important ... It furnishes the explanation of all those elementary manifestations of nature which involve time.”
Sophus Lie, Leipziger Berichte, 47 (1805)
Equations containing one independent and one dependent variable, as well as the derivative of the dependent variable with respect to the independent variable, are known as first-order ordinary differential equations (or first-order ODEs for short). An example of a first-order ODE where the independent variable is x and the dependent variable y is:
The right hand side of this differential equation depends on t and y and, consequently, is known as a non-autonomous differential equation. When the right hand side only depends on the dependent variable, the equation is autonomous. In this Chapter, we develop ...
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