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Slope field
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<h2 id="definition">Definition</h2> <h3>Standard case</h3> <p>The slope field can be defined for the following type of differential equations </p> y ′ = f ( x , y ) , {\displaystyle y'=f(x,y),} <p>which can be interpreted geometrically as giving the <a href="/facts/Slope/a2L3k0j5">slope</a> of the <a href="/facts/Tangent/sF0jH3EI">tangent</a> to the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of the differential equation's solution (<i><a href="/facts/Integral_curve/wGbVcymt">integral curve</a></i>) at each point (<i>x</i>, <i>y</i>) as a function of the point coordinates.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>It can be viewed as a creative way to plot a real-valued function of two real variables f ( x , y ) {\displaystyle f(x,y)} as a planar picture. Specifically, for a given pair x , y {\displaystyle x,y} , a vector with the components [ 1 , f ( x , y ) ] {\displaystyle [1,f(x,y)]} is drawn at the point x , y {\displaystyle x,y} on the x , y {\displaystyle x,y} -plane. Sometimes, the vector [ 1 , f ( x , y ) ] {\displaystyle [1,f(x,y)]} is normalized to make the plot better looking for a human eye. A set of pairs x , y {\displaystyle x,y} making a rectangular grid is typically used for the drawing. </p><p>An <a href="/facts/Isocline/zVoS7t2H">isocline</a> (a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form y ′ = f ( x , y ) {\displaystyle y'=f(x,y)} , the isocline is a line in the x , y {\displaystyle x,y} -plane obtained by setting f ( x , y ) {\displaystyle f(x,y)} equal to a constant. </p> <h3>General case of a system of differential equations</h3> <p>Given a system of differential equations, </p> d x 1 d t = f 1 ( t , x 1 , x 2 , … , x n ) d x 2 d t = f 2 ( t , x 1 , x 2 , … , x n ) ⋮ d x n d t = f n ( t , x 1 , x 2 , … , x n ) {\displaystyle {\begin{aligned}{\frac {dx_{1}}{dt}}&=f_{1}(t,x_{1},x_{2},\ldots ,x_{n})\\{\frac {dx_{2}}{dt}}&=f_{2}(t,x_{1},x_{2},\ldots ,x_{n})\\&\;\;\vdots \\{\frac {dx_{n}}{dt}}&=f_{n}(t,x_{1},x_{2},\ldots ,x_{n})\end{aligned}}} <p>the slope field is an array of slope marks in the <a href="/facts/Phase_space/qPe4Fq57">phase space</a> (in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ODE</a>, as seen to the right). Each slope mark is centered at a point ( t , x 1 , x 2 , … , x n ) {\displaystyle (t,x_{1},x_{2},\ldots ,x_{n})} and is parallel to the vector </p> ( 1 f 1 ( t , x 1 , x 2 , … , x n ) f 2 ( t , x 1 , x 2 , … , x n ) ⋮ f n ( t , x 1 , x 2 , … , x n ) ) . {\displaystyle {\begin{pmatrix}1\\f_{1}(t,x_{1},x_{2},\ldots ,x_{n})\\f_{2}(t,x_{1},x_{2},\ldots ,x_{n})\\\vdots \\f_{n}(t,x_{1},x_{2},\ldots ,x_{n})\end{pmatrix}}.} <p>The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen such that the points ( t , x 1 , x 2 , … , x n ) {\displaystyle (t,x_{1},x_{2},\ldots ,x_{n})} make a uniform grid. The standard case, described above, represents n = 1 {\displaystyle n=1} . The general case of the slope field for systems of differential equations is not easy to visualize for n > 2 {\displaystyle n>2} . </p> <h2 id="general-application">General application</h2> <p>With computers, complicated slope fields can be quickly made without tedium, and so an only recently practical application is to use them merely to get the feel for what a solution should be before an explicit general solution is sought. Of course, computers can also just solve for one, if it exists. </p><p>If there is no explicit general solution, computers can use slope fields (even if they aren’t shown) to numerically find graphical solutions. Examples of such routines are <a href="/facts/Euler%2527s_method/7yF5B4z9">Euler's method</a>, or better, the <a href="/facts/Runge%25E2%2580%2593Kutta_methods/YRjbyHpY">Runge–Kutta methods</a>. </p> <h2 id="software-for-plotting-slope-fields">Software for plotting slope fields</h2> <p>Different software packages can plot slope fields. </p> <h3>Direction field code in <a href="/facts/GNU_Octave/vVVq5evM">GNU Octave</a>/<a href="/facts/MATLAB/qPjLISCk">MATLAB</a></h3> funn = @(x, y)y-x; % function f(x, y) = y-x [x, y] = meshgrid(-5:0.5:5); % intervals for x and y slopes = funn(x, y); % matrix of slope values dy = slopes ./ sqrt(1 + slopes.^2); % normalize the line element... dx = ones(length(dy)) ./ sqrt(1 + slopes.^2); % ...magnitudes for dy and dx h = quiver(x, y, dx, dy, 0.5); % plot the direction field set(h, "maxheadsize", 0.1); % alter head size <h3>Example code for <a href="/facts/Maxima_(software)/3VSSJmY7">Maxima</a></h3> /* field for y'=xy (click on a point to get an integral curve). Plotdf requires Xmaxima */ plotdf( x*y, [x,-2,2], [y,-2,2]); <h3>Example code for <a href="/facts/Mathematica/dRwoGFG2">Mathematica</a></h3> (* field for y'=xy *) VectorPlot[{1,x*y-5x},{x,-2,2},{y,-2,2}] <h3>Example code for <a href="/facts/SageMath/5mzh5TlB">SageMath</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></h3> var('x,y') plot_slope_field(x*y, (x,0,-5), (y,-5,5)) <h2 id="examples">Examples</h2> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Examples_of_differential_equations/IFQTwX5o">Examples of differential equations</a></li> <li><a href="/facts/Vector_field/ccFNuPPp">Vector field</a></li> <li><a href="/facts/Laplace_transform_applied_to_differential_equations/Loe64I7j">Laplace transform applied to differential equations</a></li> <li><a href="/facts/List_of_dynamical_systems_and_differential_equations_topics/PRxKN338">List of dynamical systems and differential equations topics</a></li> <li><a href="/facts/Qualitative_theory_of_differential_equations/g9gwdyMo">Qualitative theory of differential equations</a></li></ul> <ul><li>Blanchard, Paul; <a href="/facts/Robert_L._Devaney/1zwsJpu6">Devaney, Robert L.</a>; and Hall, Glen R. (2002). <i>Differential Equations</i> (2nd ed.). Brooks/Cole: Thompson Learning. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-534-38514-1</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/SlopeField.html">"Slope field"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://www.math.psu.edu/cao/DFD/Dir.html">Slope field plotter (Java)</a></li> <li><a href="https://homepages.bluffton.edu/~nesterd/apps/slopefields.html">Slope field plotter (JavaScript)</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Boyce, William (2001). Elementary differential equations and boundary value problems (7 ed.). Wiley. p. 3. ISBN 9780471319993. <a href="9780471319993" target="_blank">9780471319993</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Vladimir A. Dobrushkin (2014). Applied Differential Equations: The Primary Course. CRC Press. p. 13. ISBN 978-1-4987-2835-5. <a href="978-1-4987-2835-5" target="_blank">978-1-4987-2835-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Andrei D. Polyanin; Alexander V. Manzhirov (2006). Handbook of Mathematics for Engineers and Scientists. CRC Press. p. 453. ISBN 978-1-58488-502-3. <a href="978-1-58488-502-3" target="_blank">978-1-58488-502-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Plotting fields — Sage 9.4 Reference Manual: 2D Graphics". <a href="https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot_field.html" target="_blank">https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot_field.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>