Informal Geometrical Description
Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.
The idea is that while the curve is initially unknown, its starting point, which we denote by is known (see the picture on top right). Then, from the differential equation, the slope to the curve at can be computed, and so, the tangent line.
Take a small step along that tangent line up to a point Along this small step, the slope does not change too much, so will be close to the curve. If we pretend that is still on the curve, the same reasoning as for the point above can be used. After several steps, a polygonal curve is computed. In general, this curve does not diverge too far from the original unknown curve, and the error between the two curves can be made small if the step size is small enough and the interval of computation is finite.
Read more about this topic: Euler Method
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