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Summary: §1.4: EULER'S METHOD
A `typical' ODE is not separable, and in fact has no closed form
solution in terms of elementary functions. The slope field is the only
way to study it. What shall we do if we need numerical answers?
Suppose we want a solution Y (t) to the ODE
Y = f(t, Y ) through the point (t0, Y0).
Geometrically, what we will do is just follow the slope field vec-
tors for a very small distance, and then adjust our direction as the
vector changes, and repeat the process. How can we write this in
equations?
At the point (t0, Y0) we know the derivative, it is just f(t0, Y0), so
we have a tangent line approximation to Y (t) at (t0, Y0):
Y - Y0 = f(t0, Y0)(t - t0) or Y = f(t0, Y0)(t - t0) + Y0.
So
Y f(t0, Y0)(t - t0) + Y0.
The problem is that the approximately equal is only true for t very
close to t0.
Euler's solution to this problem is very elegant in concept. Use the
linear approximation as above to compute some value Y1 = Y (t1),
where t1 = t0 + H for some very small H. Then repeat this pro-
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