## Abstract

Given a first-order linear differential equation and its solution, \(y(x_0),\) at a point \(x=x_0,\) Runge–Kutta procedure is used to estimate \(y(x_0+\Delta ).\) Runge–Kutta procedure is the least accurate when it uses only one step for the entire move. And indeed, as noted later, the single-step process does yield grossly inaccurate results. The two-step process—see (10.12) and (10.13)—improves the results only slightly. But the four-step effort—see (10.14)–(10.17)—does much better. It reduces the error to about \(\left( \frac{1}{50}\right) \)th of that for the one-step process. Estimates from a ten-step Runge–Kutta process are recorded in Table 10.1. These estimates—being in error only by \(\left( 100\times \frac{0.0025}{22.17}\right) =0.0113\% \)—are highly accurate.

$$\begin{aligned} \frac{\mathrm{d}y(x)}{\mathrm{d}x}= & {} F(x, y)~~, \end{aligned}$$

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