Differential Equations and Linear Algebra, 4th Edition

1 First-Order Differential Equations

1.1 Differential Equations and Mathematical Models

The laws of the universe are written in the language of mathematics. Algebra is sufficient to solve many static problems, but the most interesting natural phenomena involve change and are described by equations that relate changing quantities.

Because the derivative $dx/dt=f\text{′}(t)$ of the function f is the rate at which the quantity $x=f(t)$ is changing with respect to the independent variable t, it is natural that equations involving derivatives are frequently used to describe the changing universe. An equation relating an unknown function and one or more of its derivatives is called a differential equation.

Example 1

The differential equation

$${\displaystyle \frac{dx}{dt}}=x^2+t^2$$

involves ...