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of a scalar function. Solutions to a slope field are functions drawn as solid curves. A slope field shows the slope of a differential equation at certain vertical and horizontal intervals on the x-y plane, and can be used to determine the approximate tangent slope at a point on a curve, where the
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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
1090:{\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}}}
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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
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1425:{\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}}.}
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The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen such that the points
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640:(a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form
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1874:/* field for y'=xy (click on a point to get an integral curve). Plotdf requires Xmaxima */ plotdf( x*y, , );
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1532:. The general case of the slope field for systems of differential equations is not easy to visualize for
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It can be viewed as a creative way to plot a real-valued function of two real variables
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The slope field can be defined for the following type of differential equations
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is normalized to make the plot better looking for a human eye. A set of pairs
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Isoclines (blue), slope field (black), and some solution curves (red)
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make a uniform grid. The standard case, described above, represents
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298:) is a graphical representation of the solutions to a first-order
18:
1108:, as seen to the right). Each slope mark is centered at a point
2013:
Elementary differential equations and boundary value problems
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var('x,y') plot_slope_field(x*y, (x,-2,2), (y,-2,2))
633:
making a rectangular grid is typically used for the drawing.
16:
Visual representation of solutions to a differential equation
1987:
List of dynamical systems and differential equations topics
2091:"Plotting fields — Sage 9.4 Reference Manual: 2D Graphics"
279:{\displaystyle {\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-2x-4}
147:{\displaystyle {\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-2x+4}
210:{\displaystyle {\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-2x}
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which can be interpreted geometrically as giving the
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curve is some solution to the differential equation.
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as a planar picture. Specifically, for a given pair
2063:Andrei D. Polyanin; Alexander V. Manzhirov (2006).
1982:
Laplace transform applied to differential equations
1586:Different software packages can plot slope fields.
2039:Applied Differential Equations: The Primary Course
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1172:{\displaystyle (t,x_{1},x_{2},\ldots ,x_{n})}
8:
1992:Qualitative theory of differential equations
2110:(2nd ed.). Brooks/Cole: Thompson Learning.
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1972:Examples of differential equations
1582:Software for plotting slope fields
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2152:Slope field plotter (JavaScript)
2015:(7 ed.). Wiley. p. 3.
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686:, the isocline is a line in the
2036:Vladimir A. Dobrushkin (2014).
1735:% normalize the line element...
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1789:% ...magnitudes for dy and dx
2106:; and Hall, Glen R. (2002).
712:-plane obtained by setting
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2147:Slope field plotter (Java)
2069:. CRC Press. p. 453.
1834:% plot the direction field
361:{\displaystyle y'=f(x,y),}
2042:. CRC Press. p. 13.
1887:(* field for y'=xy *)
679:{\displaystyle y'=f(x,y)}
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1696:% matrix of slope values
1636:% function f(x, y) = y-x
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1590:Direction field code in
2011:Boyce, William (2001).
1849:"maxheadsize"
1669:% intervals for x and y
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2108:Differential Equations
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1576:Runge–Kutta methods
1562:General application
1525:{\displaystyle n=1}
705:{\displaystyle x,y}
626:{\displaystyle x,y}
553:{\displaystyle x,y}
527:{\displaystyle x,y}
454:{\displaystyle x,y}
23:The slope field of
2130:Weisstein, Eric W.
2104:Devaney, Robert L.
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2102:Blanchard, Paul;
2076:978-1-58488-502-3
2049:978-1-4987-2835-5
1896:Example code for
1878:Example code for
1867:Example code for
1861:% alter head size
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600:{\displaystyle }
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664:
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312:Standard case
311:
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754:
635:
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394:
390:
384:
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315:
295:
291:
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1936:Slope field
1880:Mathematica
1102:phase space
292:slope field
2161:Categories
1998:References
1890:VectorPlot
1592:GNU Octave
307:Definition
2138:MathWorld
1478:…
1393:…
1341:⋮
1318:…
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271:−
262:−
242:−
199:−
179:−
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2167:Calculus
1966:See also
1925:y' = x/y
1920:Examples
1898:SageMath
1642:meshgrid
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652:′
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377:tangent
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1869:Maxima
1798:quiver
1777:slopes
1750:length
1723:slopes
1705:slopes
1672:slopes
1596:MATLAB
217:, and
381:graph
373:slope
2112:ISBN
2071:ISBN
2044:ISBN
2017:ISBN
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