University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2022
Views:

173

Pages:

2
Author:

Waldx1w

Discussion: Maple and Direction Fields The purpose of this document is to look at how Maple defines a differential equation and plots its direction field. We will define a differential equation, then we will assign that DE to a variable in Maple that will make it easier to do things with it later. First, we will define the derivative in terms of the diff command: Math notation: y 0 (t) y 0 (x) y 00 (t) y 000 (t) Maple notation: diff(y(t),t) diff(y(x),x) diff(y(t),t$2) diff(y(t),t$3) And so on. Further, if y appears in the differential equation, be sure to write it in Maple as either y(t) or y(x) so that Maple knows this is a function (and not a variable). EXAMPLE 1: (See page 41 of Boyce and Diprima) Plot the direction field for y0 = x2 1 − y2 First, define the differential equation and assign it to a variable (in this case, diffeqn ). In the first example, we treat y(x) as an expression in x: diffeqn:=diff(y(x),x)=x^2/(1-(y(x))^2); And now we can plot the direction field using the DEplot command. In order to use this command, we first have to tell Maple to load in the package of commands called DEtools. Here are the two commands: with(DEtools): DEplot(diffeqn,y(x),x=-4..4,y=-4..4); You’ll notice that we needed to tell Maple (1) the differential equation, (2) What the independent and dependent variables are (the statement y(x) does this), and (3) the window size that we wanted to plot. We’ll now take a look at some optional commands. Try redoing the last line, and make a note at how the plot changes: DEplot(diffeqn,y(x),x=-4..4,y=-4..4, title="First Plot",arrows=LARGE); DEplot(diffeqn,y(x),x=-4..4,y=-4..4, dirgrid=[30,30]); 1 We can use the same command to plot some sample solution curves along with the direction field. To do this, we will have to include some initial values. EXAMPLE 2: Plot the direction field for the given differential equation, together with the three sample solutions through (1, 2), (1, 0) and (1, 1), if ty 0 + 2y = 4t2 First, define the differential equation, then plot: diffeqn:=t*diff(y(t),t)+2*y(t)=4*t^2; DEplot(diffeqn,y(t),t=0.01..2,[[y(1)=2],[y(1)=0],[y(1)=1]],y=-2..4); Output from Maple 2

Discussion Maple and Direction Fields

Related Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock