Differential Equations and Linear Algebra, 4th Edition

2.4 Numerical Approximation: Euler’s Method

It is the exception rather than the rule when a differential equation of the general form

$${\displaystyle \frac{dy}{dx}}=f(x,y)$$

can be solved exactly and explicitly by elementary methods like those discussed in Chapter 1. For example, consider the simple equation

$${\displaystyle \frac{dy}{dx}}=e^{-x^2}\text{.}$$ (1)

A solution of Eq. (1) is simply an antiderivative of $e^{-x^2}\text{.}$ But it is known that every antiderivative of $f(x)=e^{-x^2}$ is a nonelementary function—one that cannot be expressed as a finite combination of the familiar functions of elementary calculus. Hence no particular solution of Eq. (1) is finitely expressible in terms of elementary functions. Any attempt ...