1.6.4 Numerical Differential Equations

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

This generates a numerical solution to the equation with . The result is given in terms of an InterpolatingFunction.

In[1]:=  NDSolve[{y'[x] y[x], y[0] 1}, y, {x, 0, 2}] Out[1]=

Here is the value of .

In[2]:=  y[1.5] /. % 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 y[x]
InterpolatingFunction[data][x]	evaluate an interpolated function at the point x
Plot[Evaluate[y[x] /. solution], {x, , }]
	plot the solution to a differential equation

NDSolve[eqn, u, {x, , }, {t, , }, ... ]
	solve a partial differential equation