Slope field

A directional field is used for graphic determination of approximate solutions of a differential equation.

Mathematical Description

A direction field of an ( explicit) differential equation ( first order) is formed by assigning each point in the plane a vector with slope. This specifies the direction in which the graph of possible solutions of the differential equation, passing through the point of run.

Practically, this means that in a coordinate system arbitrary points are chosen and to the slope is calculated by substituting into the differential equation. ( For the derivative of y, corresponds exactly to the slope of the function. )

For the equation of the tangent individual pieces of length:

Example

The differential equation has at all points, the slope is 0, as this is given by. At the point she is, in point then. With a sufficient number of points you get a direction field are some of the possible solutions, at least initially visible.

Octave script for direction field

The script for GNU Octave draw a direction field for a differential equation of the first degree.

Function field direction ( the like ) % The like is the first derivative of x and t iA a function of x and t   % Cut and distance between the vectors x = -5:1:5; t = -5: .5:5;   for x_n = 1: length (x )    for t_n = 1: length (t )      len = sqrt (DGL (x ( x_n ), t ( t_n )) ^ 2 1); % Length of the vector for normalization      dt ( x_n, t_n ) = 1 / len; % Length of the vector along the abscissa      dx ( x_n, t_n ) = the like (x ( x_n ), t ( t_n )) / len; % Length of the vector along the ordinate    end end   quiver ( t, x, dx dt, ' r '); % Draw vectors   print (' field.svg ', ' - DSVG ') % export plot as svg file % print (' field.png ', ' - dpng ') % alternatively as a png file Save As ' Richtungsfeld.m '. Calling the script for the differential equation as follows:

The like = @ (x, t) x - t% function definition Field direction ( the like ) % to the script see also

  • Isoclines: curves connecting the points with the same pitch
  • Trajectory ( Mathematics )
  • Phase space
  • Vector field
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