slope field

The slope field can be defined for the following type of differential equations

:$y\text{'}\; =\; f(x,\; y),$

which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution (integral curve) at each point (x, y) as a function of the point coordinates.{{cite book|author1=Andrei D. Polyanin|author2=Alexander V. Manzhirov|title=Handbook of Mathematics for Engineers and Scientists|url=https://books.google.com/books?id=ge6nk9W0BCcC&pg=PA453|year=2006|publisher=CRC Press|isbn=978-1-58488-502-3|page=453}}

It can be viewed as a creative way to plot a real-valued function of two real variables $f(x,y)$ as a planar picture. Specifically, for a given pair $x,y$, a vector with the components $[1,\; f(x,y)]$ is drawn at the point $x,y$ on the $x,y$-plane. Sometimes, the vector $[1,\; f(x,y)]$ is normalized to make the plot better looking for a human eye. A set of pairs $x,y$ making a rectangular grid is typically used for the drawing.

An isocline (a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form $y\text{'}=f(x,y)$, the isocline is a line in the $x,y$-plane obtained by setting $f(x,y)$ equal to a constant.

Given a system of differential equations,

:$\backslash begin\{align\}$

\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{align}

the slope field is an array of slope marks in the phase space (in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear ODE, as seen to the right). Each slope mark is centered at a point $(t,x\_1,x\_2,\backslash ldots,x\_n)$ and is parallel to the vector

:$\backslash begin\{pmatrix\}\; 1\; \backslash \backslash \; f\_1(t,x\_1,x\_2,\backslash ldots,x\_n)\; \backslash \backslash \; f\_2(t,x\_1,x\_2,\backslash ldots,x\_n)\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; f\_n(t,x\_1,x\_2,\backslash ldots,x\_n)\; \backslash end\{pmatrix\}.$

The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen such that the points $(t,x\_1,x\_2,\backslash ldots,x\_n)$ make a uniform grid. The standard case, described above, represents $n=1$. The general case of the slope field for systems of differential equations is not easy to visualize for $n>2$.

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.

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 Euler's method, or better, the Runge–Kutta methods.

Different software packages can plot slope fields.

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

/* 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]);

(* field for y'=xy *)

VectorPlot[{1,x*y},{x,-2,2},{y,-2,2}]

var('x,y')

plot_slope_field(x*y, (x,0,-5), (y,-5,5))

Image:Slope_field_1.svg|Slope field

Image:Slope_field_with_integral_curves_1.svg|Integral curves

image:Isocline_3.png|Isoclines (blue), slope field (black), and some solution curves (red)

• Examples of differential equations
• Vector field
• Laplace transform applied to differential equations
• List of dynamical systems and differential equations topics
• Qualitative theory of differential equations

{{Reflist}}

• Blanchard, Paul; Devaney, Robert L.; and Hall, Glen R. (2002). Differential Equations (2nd ed.). Brooks/Cole: Thompson Learning. {{ISBN|0-534-38514-1}}

• {{MathWorld |title = Slope field |urlname = SlopeField}}
• [http://www.math.psu.edu/cao/DFD/Dir.html Slope field plotter (Java)]
• [https://homepages.bluffton.edu/~nesterd/apps/slopefields.html Slope field plotter (JavaScript)]

Category:Calculus

Category:Differential equations

Category:Articles with example MATLAB/Octave code

Category:Plots (graphics)