SolveODE• SolveODE[eqns, y, {x, xmin, xmax}] finds a numerical solution to the ordinary differential equations eqns for the function y with the independent variable x in the range xmin to xmax.
• SolveODE[eqns, y, {x, xmin, xmax}, {t, tmin, tmax}] finds a numerical solution to the partial differential equations eqns.
• SolveODE[eqns, { ,  , ... }, {x, xmin, xmax}] finds numerical solutions for the functions  . • SolveODE gives results in terms of InterpolatingFunction objects.
• SolveODE[eqns, y[x], {x, xmin, xmax}] gives solutions for y[x] rather than for the function y itself.
• SolveODE solves a wide range of ordinary differential equations, and some partial differential equations.
• In ordinary differential equations, the functions  must depend only on the single variable x. In partial differential equations, they may depend on more than one variable.
• The differential equations must contain enough initial or boundary conditions to determine the solutions for the  completely.
• Initial and boundary conditions are typically stated in form y[ ]   , y'[ ]  , etc., but may consist of more complicated equations.
• The point  that appears in the initial or boundary conditions need not lie in the range xmin to xmax over which the solution is sought.
• The differential equations in SolveODE can involve complex numbers.
• See also: D, Integrate, ND, NIntegrate, SolveEquation.
Examples Using InstantCalculatorsHere are the InstantCalculators for the SolveODE function. Enter the parameters for your calculation and click Calculate to see the result.
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Entering Commands DirectlyYou can paste a template for this command via the Text Input button on the SolveODE Function Controller. This command finds a numerical approximation to a function that is equal to its first derivative at each point x between  and  , and that has the value  when x is  . SolveODE returns an InterpolatingFunction object.
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This can be plotted by making a function from the InterpolatingFunction object.
Here is another example. This finds a numerical approximation to a function whose square is equal to its first derivative.
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The warning is appropriate, since this function goes to  as it approaches 1 along the horizontal axis.
Here is how to see a graph of an approximation to the function which is the reciprocal of its derivative.
Not too surprisingly, this plot looks very much like the square root function.
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Solving systems of equations works similarly. For systems of two equations a so-called phase plot is often a good way to visualize the solution. Here is a phase plot that describes the motion of a weakly damped pendulum.
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Clear the function definition.
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