Differential Equations and Linear Algebra, 4th Edition

An Improvement in Euler’s Method

As Fig. 2.5.2 shows, Euler’s method is rather unsymmetrical. It uses the predicted slope $k=f(x_n,y_n)$ of the graph of the solution at the left-hand endpoint of the interval $[x_n,x_n+h]$ as if it were the actual slope of the solution over that entire interval. We now turn our attention to a way in which increased accuracy can easily be obtained; it is known as the improved Euler method.

Given the initial value problem

$${\displaystyle \frac{dy}{dx}}=f(x,y),y(x_0)=y_0,$$ (4)

suppose that after carrying out n steps with step size h we have computed the approximation $y_n$ to the actual value $y(x_n)$ of the solution at $x_n=x_0+nh\text{.}$ We can use the Euler method to obtain a first estimate—which we now call $u_{n+1}$ rather than $y_{n+1}$—of the value of the solution at ...