Direction field

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

The solutions of a differential equation of the first order of a scalar function y (x) can be drawn in a 2-dimensional space with x in the horizontal and y in the vertical direction. Possible solutions are functions y (x), which are drawn by curves. Sometimes it is difficult to solve the differential equation analytically. Then you can, however, the tangents of the function curves z. B. draw on a regular grid. The tangents touch the functions at the grid points.

Mathematical description

A direction field of a differential equation (first order) is created by assigning a vector with a slope to each point in the plane . This indicates the direction in which the graphs of possible solutions of the differential equation that go through the point run. ${\displaystyle y'(x)=F(x,y(x))}$ ${\displaystyle (x,y)}$ ${\displaystyle F(x,y)}$ ${\displaystyle (x,y)}$

In practical terms, this means that arbitrary points can be selected in a coordinate system and that the slope is calculated by inserting it into the differential equation. (Because the derivative of corresponds to the slope of the function.) ${\displaystyle P(x,y)}$ ${\displaystyle y'}$ ${\displaystyle y}$

The equation of the individual tangent segments of the length is : ${\displaystyle y'(x)=F(x,y(x))}$ ${\displaystyle l}$

${\displaystyle {\vec {v}}(x,y;\lambda )={\begin{pmatrix}x\\y\end{pmatrix}}+{\frac {\lambda }{\sqrt {1+F^{2}(x,y)}}}{\begin{pmatrix}1\\F(x,y)\end{pmatrix}}\quad {\text{mit}}\quad \lambda =\left[-{\frac {l}{2}},{\frac {l}{2}}\right]}$

The isoclines , given by the equation , i.e. the lines with the same gradient, are often helpful in the graphic representation . ${\displaystyle F(x,y)=\mathrm {const} }$

example

Direction field for ${\displaystyle y'=y-x}$

The differential equation has the slope value 0 in all points , since this is given by . In the point it is then in the point . With enough points you get a directional field in which multitudes of possible solutions can be rudimentarily made visible through their functional tangents. ${\displaystyle y'(x)=y(x)-x}$ ${\displaystyle (C,C)}$ ${\displaystyle y-x=C-C=0}$ ${\displaystyle P_{1}(x,y)=(1,2)}$ ${\displaystyle 2-1=1}$ ${\displaystyle P_{2}(x,y)=(-4,2)}$ ${\displaystyle 2-(-4)=6}$

Octave script for direction field

The script direction field.m is written for GNU Octave and draws a direction field for DGL , a differential equation of the first degree. ${\displaystyle {\dot {y}}(x)=y(x)-x}$

% Inhalt des Files ''richtungsfeld.m''
function richtungsfeld(dgl)
% dgl ist die erste Ableitung von y nach x und ist i.A. eine Funktion von x und y
% Ausschnitt und Abstand zwischen den Vektoren
y = -5:.5:5; x = -5:.5:5;
for y_n = 1:length(y)
 for x_n = 1:length(x)
   len = sqrt( dgl(y(y_n), x(x_n))^2 + 1 ); % Länge des Vektors für Normierung
   dx(y_n,x_n) = 1 / len;                   % Länge des Vektors entlang der Abszisse
   dy(y_n,x_n) = dgl(y(y_n), x(x_n)) / len; % Länge des Vektors entlang der Ordinate
 end
end 
h=quiver(x, y, dx, dy,0.5,"r","linewidth",1); % Vektoren zeichnen 
set (h, "maxheadsize", 0.1); 
xlabel ("x");
ylabel("y");
print('field.svg', '-dsvg')  % Plot als svg-Datei exportieren
% Ende des Files

- Now call up the file as follows within an Octave session:

source("richtungsfeld.m")
dgl = @(y, x) y-x   % Funktionsdefinition 
richtungsfeld(dgl)

See also

• Trajectory (mathematics)
• Phase space
• Vector field

literature

• W. Walter: Ordinary Differential Equations: An Introduction. 7th edition. Springer, Berlin 2000, ISBN 3-540-67642-2
• F. Reinhardt, H. Soeder: dtv-Atlas Mathematik. Volume 2. 11th edition. Deutscher Taschenbuch Verlag, 1998, ISBN 3-423-03008-9