Plotting Slope Fields and Solving Differential Equations on Maple

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Plotting Slope Fields and Solving Differential
Equations on Maple

• Marie Bruley
• Math 4B
• Merced College

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What is a slope field?

• A slope field (or direction field) plots the
  direction of the slopes of the tangent lines to a
  family of functions at various points.
• The family of functions represents the integral
  curves that satisfy a given differential equation.

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Plotting Slope Fields

• Commands
• with (DEtools)
• dfieldplot(diff(y(x),x)2y(x)-x,y(x),x-1..5,y-
  1..5,color2y-x)

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The result!
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Solving Differential Equations

• Commands
• ode diff(y(x),x)2y(x)-x
• dsolve(ode)

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Differential equations with initial conditions

• Commands (note this follows after the output
  from the dsolve(ode) command)
• ics y(0)1
• dsolve(ode,ics)

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Plot of the solution
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Compare the slope field to the plot of the
solution
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Change the initial conditions

• For y(0)-1
• The solution is

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Compare to the slope field
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The slope field allows us to see what the
behavior of our function should be, in terms of
looking at its tangent lines. This can be
valuable for differential equations that cant be
solved explicitly.
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Definitions

• Autonomous D.E. is one where the independent
  variable (x) does not appear explicitly in the
  differential equation.
• Critical point a value c where f(c)0 in the
  autonomous D.E.
• Ex C-1 is a critical point.
• Critical points are also called stationary or
  equilibrium points.
• If C is a critical point then y(x)C is a
  constant solution (equilibrium solution).

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Attractors and Repellers