ON A HOMEWORK PROBLEM

This is intended to give you some hints for problem 5 of the WeBWorK assignment "set s03day1".

Load first the differential equations library:

> with(DEtools ):

Next we enter the differential equation:

> DE := diff(y(x),x)=y^3-x^3:

Now ask Maple to do a direction field plot of this differential equation on a suitable domain:

> DEplot( DE ,y(x), x=-3..3, y=-3..3, color=BLUE);

Warning, y is present as both a dependent variable and a name. Inconsistent specification of the dependent variable is deprecated, and will be removed in the next release.

Question: What information do you think is important in this graph? What does the picture say? In what respect is it different from the

direction field of dy/dx=y-x?

Well, one can look for example at the points where the derivative is zero. This amounts to solving for the first DE the equation

y^3=x^3, with solutions y=x. Note that these are exactly the places where the arrows in the direction field are horizontal.

The same description of the zeros holds for the direction field of the second equation. We can further look at the points where

the derivative is constant. This amounts to solving for the second DE the equation y-x=constant, which gives y=x+constant.

These are exactly the lines parallel to the line y=x. In other words the arrows in the direction field of dy/dx=y-x have the same

inclination (slope) on each line parallel to the line y=x. This is not the case for the direction field above, so we can say for sure

that it does not represent the DE dy/dx=y-x.

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Another feature of Maple is that it can draw the solutions of differential equation which pass through several points:

> DE := diff(y(x),x)=y^3-x^3:

> DEplot( DE , y(x), x=-3..3, y=-3..3,
[[y(0)=0], [y( 0)=1], [y(0)=-1]], arrows=SMALL,
dirgrid=[30, 30], stepsize=.1,
linecolor=RED, color=BLUE);

Warning, y is present as both a dependent variable and a name. Inconsistent specification of the dependent variable is deprecated, and will be removed in the next release.

Such information, together with the similar graphs for the other direction field, can also be used to distinguish

between the vector fields, and to understand some qualitative things about the diferential equations.