文档介绍：Section
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Numerical Solutions of
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If x is a function of a real variable t and f is a function of both x and t, then the equation
x˙(t) = f(x(t), t) ()
is called a first order differential equation. Solving such an equation involves more than
algebraic manipulation; indeed, although the equation itself involves three quantities, x,
x˙, and t, to find a solution we must identify a function x, defined solely in terms of
the independent variable t, which satisfies the relationship of () for all t in some
open interval. For many equations, exact solution is not possible and we have to rely on
approximations. In this chapter we will discuss techniques for finding both approximate
and, where possible, exact solutions to differential equations.
We have already seen many examples of differential equations: in Section when
we discussed finding the position of an object moving in a straight line given its velocity
function and its initial position, in Section when we discussed models for growth and
decay, in Section when we discussed the motion of a projectile, and in Section
when we considered the two-body problem. Indeed, in many ways the study of differential
equations is at the heart of calculus. To study the interaction of physical bodies in the
world is to study the ramifications of physical laws such as the law of gravitation and
Newton’s second law of motion, laws which frequently lead, as we saw in Section , to
questions involving the solution of differential equations. Newton was the first to realize
the power of calculus for solving a vast array of physical problems. The mathematicians
that followed him enlarged and refined his techniques until they began to believe that the
entire future of the universe, as well as its past, could be discerned from a knowledge of
the current positions and velocities of all physical bo