Slope field

In mathematics, a slope field is a graphical tool to qualitively visualize, or aid in numerical approximation of, solutions to differential equations.

Definition

 

This is a sample slope field of dy/dt=2t+y, with two solution curves superimposed (y(0)=-1 and y(0)=-3, for red and blue respectively). The general solution is y(t)=c(e^t)-2t-2.

Given a system of differential equations,

${\displaystyle {\frac {du}{dt}}=f(t,u,...y,z)}$ 

${\displaystyle \cdots }$ 

${\displaystyle {\frac {dy}{dt}}=j(t,u,...y,z)}$ 

${\displaystyle {\frac {dz}{dt}}=k(t,u,...y,z)}$ 

the slope field is an array of slope marks in the phase space (the preceding equations imply seven dimensions, but can be any number 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,u,...y,z) and is parallel to the vector

${\displaystyle {\begin{pmatrix}1\\f(t,u,...y,z)\\\cdots \\j(t,u,...y,z)\\k(t,u,...y,z)\end{pmatrix}}}$ .

The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen as (t,u,...y,z)=(aΔt, bΔu, ... eΔy, fΔz) for arbitrary (but usually equal) Δt, Δu, ... Δy, and Δz, and for all integers a, b, ... e, and f that produce points within the chosen t, u, ... y, and z intervals. The length of the slope marks is usually uniform throughout, and unitary or no greater than the least of Δt, Δu, ... Δy, and Δz.

General application

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.

See also

• Examples of differential equations
• Differential equations of mathematical physics
• Differential equations from outside physics
• Laplace transform applied to differential equations
• List of dynamical systems and differential equation topics

External links

• Slope field from MathWorld

References

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

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