Alex au pays des chiffres by Alex Bellos

By Alex Bellos

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A car traveling at 60 mi/h (88 ft/s) skids 176 ft after its brakes are suddenly applied. Under the assumption that the braking system provides constant deceleration, what is that deceleration? For how long does the skid continue? 31. The skid marks made by an automobile indicated that its brakes were fully applied for a distance of 75 m before it came to a stop. The car in question is known to have a constant deceleration of 20 m/s2 under these conditions. How fast—in km/h—was the car traveling when the brakes were first applied?

We might think of integrating both sides in (1) with respect to x, and hence write y(x) = f (x, y(x)) d x + C. However, this approach does not lead to a solution of the differential equation, because the indicated integral involves the unknown function y(x) itself, and therefore cannot be evaluated explicitly. Actually, there exists no straightforward procedure by which a general differential equation can be solved explicitly. Indeed, the solutions of such a simple-looking differential equation as y = x 2 + y 2 cannot be expressed in terms of the ordinary elementary functions studied in calculus textbooks.

A slope field suggests visually the general shapes of solution curves of the differential equation. Through each point a solution curve should proceed in such a direction that its tangent line is nearly parallel to the nearby line segments of the slope field. Starting at any initial point (a, b), we can attempt to sketch freehand an approximate solution curve that threads its way through the slope field, following the visible line segments as closely as possible. Example 2 Solution Construct a slope field for the differential equation y = x − y and use it to sketch an approximate solution curve that passes through the point (−4, 4).

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