MATHEMATICA TUTORIAL

Introduction to Numerical Differential Equations

NDSolve[eqns,y,{x,x_min,x_max}]	solve numerically for the function y, with the independent variable x in the range to
NDSolve[eqns,{y₁,y₂,...},{x,x_min,x_max}]
	solve a system of equations for the

Numerical solution of differential equations.

This generates a numerical solution to the equation

with

. The result is given in terms of an InterpolatingFunction.

In[1]:=

Out[1]=

Here is the value of

In[2]:=

Out[2]=

With an algebraic equation such as

, each solution for

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

, you want to get an approximation to the function

as the independent variable

varies over some range.

Mathematica represents numerical approximations to functions as InterpolatingFunction objects. These objects are functions which, when applied to a particular

, return the approximate value of

at that point. The InterpolatingFunction effectively stores a table of values for

, then interpolates this table to find an approximation to

at the particular

you request.

y[x]/.solution	use the list of rules for the function y to get values for
InterpolatingFunction[data][x]	evaluate an interpolated function at the point x
Plot[Evaluate[y[x]/.solution],{x,x_min,x_max}]
	plot the solution to a differential equation