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4.1 Generate direction field plot of a first order differential equation

Problem: Given the following non autonomous differential equation, plot the line fields which represents the solutions of the ODE.

\[ \frac {dy\left ( x\right ) }{dx}=x^2 - y \]

Direction field plot (or slope plot) shows solutions for the ODE without actually solving the ODE.

The following are the steps to generate direction field plot for \(\frac {dy}{dx}=f(x,y)\)

  1. generate 2 lists of numbers. The \(y\) list and the \(x\) list. These 2 lists are the coordinates at which a small slop line will be plotted.
  2. At each of the above coordinates \((y,x)\) evaluate \(f(x,y)\).
  3. Plot small line starting at the point \((x,y)\) and having slop of \(f(x,y)\). The length of the line is kept small. Normally it has an arrow at the end of it.

Using Matlab, the points are first generated (the \((x,y)\) coordinates) then the slope \(f(x,y)\) evaluated at each of these points, then the command quiver() is used. Next contour() command is used to plot few lines of constant slope.

In Mathematica, the command VectorPlot is used. In Maple dfieldplot is used.

Mathematica 

f[x_,y_]:= x^2-y 
ListVectorPlot[Table[{1,f[x,y]}, 
        {x,-2,2,0.1},{y,-2,2,0.1}], 
      FrameLabel->{{"y(x)",None}, 
     {"x", 
      "direction fields for y'(x)=x^2-y"} 
     }, 
      ImageSize->350, 
      LabelStyle->Directive[14]]

pict 

StreamPlot[{1,f[x,y]},{x,-2,2},{y,-2,2}, 
 FrameLabel->{{"y(x)",None}, 
  {"x","direction fields for y'(x)=x^2-y"}}, 
 ImageSize->350, 
 LabelStyle->Directive[14]]

pict

 

Matlab 

clear all; close all; 
 
%direction fields for dy/dx=xy 
f     = @(X,Y) X.^2 - Y; 
[X,Y] = meshgrid(-2:.4:2,-2:.4:2); 
YlengthOfVector = f(X,Y); 
XlengthOfVector = ones(size(YlengthOfVector)); 
 
quiver(X,Y,XlengthOfVector,YlengthOfVector); 
xlabel('x'); ylabel('y'); 
hold on; 
 
contour(X,Y,YlengthOfVector,... 
  [-5 -4 -3 -2 -1 0 1 2 3 4 5]); 
 
title(... 
'direction fields for $\frac{dy}{dx}=x^2-y$',... 
'interpreter','latex','fontsize',12)

pict

 

Maple 

restart; 
with(DEtools): with(plots): 
ode1:=diff(y(x),x)=(y(x))^2-x; 
dfieldplot(ode1,y(x),x=-6..10, y=-4..4, 
 arrows=MEDIUM, 
 title=`Line fields for an non-autonomous ODE`);

pict 

restart; 
with(DEtools): with(plots): 
ode1:=diff(y(x),x)=(y(x))^2-x: 
p1:= dfieldplot(ode1,y(x),x=-6..10, y=-6..6, 
       arrows=MEDIUM): 
sol:=dsolve({ode1,y(0)=0});

\[ y \left ( x \right ) =-{\frac {\sqrt {3}{{\rm Ai}^{(1)}\left (x\right )}+{{\rm Bi}^{(1)}\left (x\right )}}{\sqrt {3}{{\rm Ai}\left (x\right )}+{{\rm Bi}\left (x\right )}}} \]
 
p2:=plot(rhs(sol),x=-6..10, 
       color=black,thickness=3): 
display({p1,p2}, 
 title=`Line fields and solution lines 
     for an non-autonomous ODE`);

pict

 

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