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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 InstantCalculators

    Here 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 Directly

    You 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.

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    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.

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    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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