We are concerned here with the problem of solving differential equations, numerically. At first we concentrate on the so-called *initial value problem* (IVP): Find a function *y*(*t*) such that

where *f* is a known function of two variables, and *t*_{0} and *y*_{0} are known values. This is called the initial value problem because (as the notation suggests) we can view the independent variable *t* as time, and the equation as modeling a process that moves forward from some initial time *t*_{0} with initial state *y*_{0}. (Very often, *t*_{0} = 0.) The dependent variable *y*, the unknown function, may be a scalar function or, possibly, a vector function defined as

In §2.3 we developed Euler’s method for approximating solutions to initial value problems; in this chapter we will not only review Euler’s method, but we will also look at more sophisticated (and therefore, we hope, more accurate) methods for solving this type of problem. Later we will tackle the boundary value problem (BVP), which can be written as

(6.1)

(6.2)

(6.3)

Here the unknown function is ...

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