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