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Introduction to Numerical Differential Equations

solve numerically for the function y, with the independent variable x in the range xmin to xmax
solve a system of equations for the yi

Numerical solution of differential equations.

This generates a numerical solution to the equation y (x)=y (x) with 0<x<2. The result is given in terms of an InterpolatingFunction.
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Here is the value of y (1.5).
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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.
y[x]/.solutionuse the list of rules for the function y to get values for y[x]
InterpolatingFunction[data][x]evaluate an interpolated function at the point x
plot the solution to a differential equation

Using results from NDSolve.

This solves a system of two coupled differential equations.
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Here is the value of z[2] found from the solution.
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Here is a plot of the solution for z[x] found on line 3. Plot is discussed in "Basic Plotting".
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solve a partial differential equation

Numerical solution of partial differential equations.