AP Calculus Problem #7

The problem

Consider the differential equation .

(a)

On the axes provided, sketch a slope field for the differential equation at the eleven indicated points.

> plot([
[-1,-1],[0,-1],[1,-1],
[-1,0],[0,0],[1,0],
[-2,1],[-1,1],[0,1],[1,1],[2,1]
],style=point,symbol=circle,symbolsize=20,view=[-3..3,-2..2],color=black);

(b)

Use the slope field to explain why a solution could not have the graph shown below:

> plot(x^2,x=-1..1,view=[-2..2,-2..2],scaling=constrained,ytickmarks=3,thickness=3);

(c)

Find the particular solution to the differential equation with the initial condition .

(d)

Find the range of the solution found in part (c).

Some Maple syntax

We are going to want to use the differential equations package.

> with(DEtools):

We take an example from Apostol of a first order separable differential equation: exercise 6 page 347:

> DE:=diff(y(x),x)=x*y(x)/(x-1);

> DEplot(DE,y(x),x=-2..2.5,[[y(0)=0],[y(-1)=2],[y(0.5)=-1]],
colour=wheat,linecolor=[red,green,magenta],arrows=large);

Note the syntax: first the differential equation, then a range for the independent variable x, then a list of "initial conditions", and options.

The differential equation in the AP problem and the one in the example above are both called separable , and as such they are rather simple to solve. You should be aware, however, that Maple does have a routine to solve many differential equations: dsolve .

> dsolve(DE);

> f:=x->C*exp(x)*(x-1);

> with(plots):

Warning, the name changecoords has been redefined

Next we look at an animation of different solutions.

> animate(f(x),x=-3..2,C=-4..4,thickness=3);

Play the animation above.

Note that all curves pass through the point [1,0]. This is clear from the functional form of the solution.

Your task

Use Maple to make a slope field for the differential equation of the AP problem. Include several solution curves, including the one from (c). Investigate the solution curves. Try to make general observations. There must something interesting to say about . Write up your observations.