With an algebraic equation such as
x2+3x+1=0, each solution for
x is simply a single number. For a differential equation, however, the solution is a
function, rather than a single number. For example, in the equation
y
(x)=y (x), you want to get an approximation to the function
y (x) as the independent variable
x varies over some range.
Mathematica represents numerical approximations to functions as
InterpolatingFunction objects. These objects are functions which, when applied to a particular
x, return the approximate value of
y (x) at that point. The
InterpolatingFunction effectively stores a table of values for
y (xi), then interpolates this table to find an approximation to
y (x) at the particular
x you request.
