Cfrgtkky

In earlier thread Jake provided some code whom successfully draws the following differential equations in the range [0; 1]

dy/dx=2*x

dy/dx=x*sqrt(x)

See: How to draw slope fields with all the possible solution curves in latex

I have not been able to determine how to change the range to [-1; 1] for both dimensions. My attempt

xmin=-1.1, xmax=1.1, % Axis limits

ymin=-1.1, ymax=1.1,

domain=-1:1, y domain=-1:1,

Also, I had some trouble with the notation for a differential equation that consists of y, example (dy/dx=x^2+y^2-1).

My attempt:

declare function={f(x) = x^2 + f(x)^2 - 1;}

Using PGFPlotstable, you can use a naive numerical integration scheme to find the function directly within LaTeX:

documentclass{article}

usepackage{pgfplots, pgfplotstable}

pgfplotsset{compat=1.8}



usepackage{amsmath}



pgfplotstableset{

    create on use/x/.style={

        create col/expr={

            pgfplotstablerow/201*2-1

        }

    },

    create on use/y/.style={

        create col/expr accum={

            pgfmathaccuma+(2/201)*(abs(pgfmathaccuma^2)+abs(thisrow{x}^2)-1)

        }{0.6}

    }

}





pgfplotstablenew{201}loadedtable



begin{document}

begin{tikzpicture}

begin{axis}[

    view={0}{90},

    domain=-1:1,

    y domain=-1:1,

    xmax=1, ymax=1,

    samples=21

]

addplot3 [gray, quiver={u={1}, v={x^2+y^2-1}, scale arrows=0.075, every arrow/.append style={-latex}}] (x,y,0);

addplot [thick, red] table [x=x, y=y] {loadedtable};

end{axis}

end{tikzpicture}



end{document} 

(Sets of) ordinary differential equations (ODE) can be numerically solved using pstODEsolve from the pst-ode Plain-TeX / LaTeX package.

Integration of the ODEs is done using the Runge-Kutta-Fehlberg method of 4th order with adaptive step size control (RKF45) during the ps2pdf conversion step.

For plotting packages other than PSTricks, such as pgfplots to be used here, LaTeX must be run twice. The first run, must go the dvips+ps2pdf route, in order to write the solution table into a text file, which is then read and transformed into a chart by your favourite plotting package and TeX engine.

The first compilation is done like this (option -dNOSAFER is necessary to allow Ghostscript to write files):

latex myfile

dvips myfile

ps2pdf -dNOSAFER myfile.ps

The second run can involve your preferred toolchain pdflatex, xelatex, latex/dvips/ps2pdf, whatever. It plots the results using pgfplots.

Alternatively, a single call to your favourite LaTeX engine with option --shell-escape is sufficient if the second code box below is used. (Under the hood, the chain of commands tex, dvips, ps2pdf -dNOSAFER is executed before the document is actually typeset.) Solution provided by Herbert.

documentclass{article}

usepackage{amsmath}

usepackage{pgfplots}

pgfplotsset{compat=1.8}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

usepackage{ifpdf}

ifpdf

IfFileExists{y0=-0.5.dat}{}{

  GenericError{}{MessageBreak%

    ===================================MessageBreak

    First, you have to runMessageBreakMessageBreak

     latex jobnameMessageBreak

     dvips jobnameMessageBreak

     ps2pdf -dNOSAFER jobname.psMessageBreakMessageBreak

    to solve the differential equation.MessageBreak

    ===================================}{}{}

}

else

usepackage{pst-ode}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%solve dy/dx=x^2 + y^2 - 1 numerically for different initial values of y in the

%interval x=[-1.1,1.1]; write resulting curves as tables with 100 output points

%into text files

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%y0=-0.5 --> y0=-0.5.dat

pstODEsolve[algebraicOutputFormat,algebraic,saveData]{%

  y0=-0.5%  %name of the output file `y0=-0.5.dat'

}{

  t | x[0]  %table format in `y0=-0.5.dat': x y

}{

  -1.1      %integration domain x_min=-1.1

}{

  1.1       %integration domain x_max=1.1

}{

  100       %number of output points

}{

  -0.5      %initial value y0(x_min)

}{

  t^2+x[0]^2-1  % right hand side of ODE, note the special notation:  x --> t, y --> x[0]

}



%y0=0.0 --> y0=0.0.dat

pstODEsolve[algebraicOutputFormat,algebraic,saveData]{y0=0.0}{t | x[0]}{-1.1}{1.1}{100}{0.0}{t^2+x[0]^2-1}



%y0=0.5 --> y0=0.5.dat

pstODEsolve[algebraicOutputFormat,algebraic,saveData]{y0=0.5}{t | x[0]}{-1.1}{1.1}{100}{0.5}{t^2+x[0]^2-1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

fi



begin{document}

IfFileExists{y0=-0.5.dat}{}{dummy textend{document}}



begin{tikzpicture}

  begin{axis}[

    axis equal image, % Unit vectors for both axes have the same length

    xmin=-1.1, xmax=1.1, % Axis limits

    ymin=-1.1, ymax=1.1,

    xtick={-1,-0.5,0,0.5,1}, ytick={-1,-0.5,0,0.5,1}, % Tick marks

    no markers,

    title={$dfrac{mathrm{d}y}{mathrm{d}x}=x^2+y^2-1$},

    view={0}{90},samples=21,domain=-1.1:1.1, y domain=-1.1:1.1, %for direction field

  ]

  addplot3 [gray, quiver={u={1}, v={x^2+y^2-1}, scale arrows=0.075, every arrow/.append style={-latex}}] (x,y,0);

  addplot table {y0=-0.5.dat};

  addplot table {y0=0.0.dat};

  addplot table {y0=0.5.dat};

  end{axis}

end{tikzpicture}



end{document}

Code for typesetting at one sweep using option --shell-escape with any LaTeX engine:

documentclass{article}

usepackage{amsmath}

usepackage{pgfplots}

pgfplotsset{compat=1.8}



usepackage{filecontents}

begin{filecontents}{xyz.tex}

input pst-ode

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%solve dy/dx=x^2 + y^2 - 1 numerically for different initial values of y in the

%interval x=[-1.1,1.1]; write resulting curves as tables with 100 output points

%into text files

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%y0=-0.5 --> y0=-0.5.dat

pstODEsolve[algebraicOutputFormat,algebraic,saveData]{y0=-0.5}{t | x[0]}{-1.1}{1.1}{100}{-0.5}{t^2+x[0]^2-1}



%y0=0.0 --> y0=0.0.dat

pstODEsolve[algebraicOutputFormat,algebraic,saveData]{y0=0.0}{t | x[0]}{-1.1}{1.1}{100}{0.0}{t^2+x[0]^2-1}



%y0=0.5 --> y0=0.5.dat

pstODEsolve[algebraicOutputFormat,algebraic,saveData]{y0=0.5}{t | x[0]}{-1.1}{1.1}{100}{0.5}{t^2+x[0]^2-1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

bye

end{filecontents}



immediatewrite18{tex xyz}

immediatewrite18{dvips xyz}

immediatewrite18{ps2pdf -dNOSAFER xyz.ps}



begin{document}



begin{tikzpicture}

  begin{axis}[

    axis equal image, % Unit vectors for both axes have the same length

    xmin=-1.1, xmax=1.1, % Axis limits

    ymin=-1.1, ymax=1.1,

    xtick={-1,-0.5,0,0.5,1}, ytick={-1,-0.5,0,0.5,1}, % Tick marks

    no markers,

    title={$dfrac{mathrm{d}y}{mathrm{d}x}=x^2+y^2-1$},

    view={0}{90},samples=21,domain=-1.1:1.1, y domain=-1.1:1.1, %for direction field

  ]

  addplot3 [gray, quiver={u={1}, v={x^2+y^2-1}, scale arrows=0.075, every arrow/.append style={-latex}}] (x,y,0);

  addplot table {y0=-0.5.dat};

  addplot table {y0=0.0.dat};

  addplot table {y0=0.5.dat};

  end{axis}

end{tikzpicture}



end{document}

MWE with Asymptote

% odeslope.tex:

%

documentclass{article}

usepackage[inline]{asymptote}

usepackage{lmodern}

begin{document}

begin{figure}

centering

begin{asy}

import graph;

import slopefield;

import fontsize;

defaultpen(fontsize(9pt));

size(200);

real dy(real x,real y) {return x^2+y^2-1;}

real xmin=-1, xmax=1;

real ymin=-0.2, ymax=1;



add(slopefield(dy,(xmin,ymin),(xmax,ymax),20,deepgreen+0.4bp,Arrow));



pair C=(0.5,0.4);

draw(curve(C,dy,(xmin,ymin),(xmax,ymax)),deepblue+1bp);



label("$C$",C,NE,UnFill);

dot(C,UnFill);



xaxis(YEquals(ymin),xmin,xmax,LeftTicks());

xaxis(YEquals(ymax),xmin,xmax);

yaxis(XEquals(xmin),ymin,ymax,RightTicks());

yaxis(XEquals(xmax),ymin,ymax);

end{asy}

caption{$frac{mathrm{d}y}{mathrm{d}x}=x^2+y^2-1$}

end{figure}

end{document}

%

% Process:

%

% pdflatex odeslope.tex

% asy odeslope-*.asy

% pdflatex odeslope.tex

A PSTricks solution, it can be run with xelatex but that takes a lot of time in difference to latex->dvips->ps2pdf

documentclass[border=10pt]{standalone}

usepackage{pst-plot,pst-ode}

begin{document}



psset{unit=3}

begin{pspicture}(-1.2,-1.2)(1.1,1.1)

psaxes[ticksize=0 4pt,axesstyle=frame,tickstyle=inner,subticks=20,

        Ox=-1,Oy=-1](-1,-1)(1,1)

psset{arrows=->,algebraic}

psVectorfield[linecolor=black!60](-0.9,-0.9)(0.9,0.9){ x^2+y^2-1 }

%y0_a=-0.5

pstODEsolve[algebraicOutputFormat]{y0_a}{t | x[0]}{-1}{1}{100}{-0.5}{t^2+x[0]^2-1}

%y0_b=0.0

pstODEsolve[algebraicOutputFormat]{y0_b}{t | x[0]}{-1}{1}{100}{0.0}{t^2+x[0]^2-1}

%y0_c=0.5

pstODEsolve[algebraicOutputFormat]{y0_c}{t | x[0]}{-1}{1}{100}{0.5}{t^2+x[0]^2-1}



psset{arrows=-,linewidth=1pt}%

listplot[linecolor=red  ]{y0_a}

listplot[linecolor=green]{y0_b}

listplot[linecolor=blue ]{y0_c}

end{pspicture}



end{document}