#14 in 2.4


> with(DEtools):
> dfieldplot(diff(y(x),x)=y*(3-x*y),y(x),x=-5..5,y=-5..5,arrows=LINE);




This is hard to look at. What does it realy mean? Notice that y' = 0
on the x-axis (y=0) and alone the hyperbola y=3/x. This may help you
to understand the behavior as x gets large. If we start at a point 
on the negative y-axis it should be clear that y goes to -infinity  
as x goes to infinity.  If we start at y(0)=0 then the limit is 0 
since y(x)=0 is a solution. (Is this solution unique? It is hard to 
tell from the plot alone.) Now for y(0) positive it is less clear 
what happens as x goes to infinity. The curves start out going up 
but then turn sharply down. Do they go to 0 in the limit or to 
some small number? Let's "zoom in".


> dfieldplot(diff(y(x),x)=y*(3-x*y),y(x),x=120..130,y=0..0.2,arrows=LINE);





I have zoomed in several times now and it seems to be going to zero. This 
makes sense since 3/x goes to zero. To be sure I solved the equation with 
dsolve:


> dsolve(diff(y(x),x)=y*(3-x*y),y(x));



                   1      1       1
                  ---- = --- x - --- + exp(-3 x) _C1
                  y(x)    3       9


For the life of me I don't know why Maple gives 1/y instead of y 
itself. But as x goes to infinity  it is clear that y(x) goes to 
0 for any C. But that does not make any sense!! From the graph it 
seemed so clear that the limit was -infinity for y(0) negative. 
What is going on? We will go over this in class, but see if you 
can figure out what is happening.  Hint: plot several solution 
curves.


BTW: We can rewrite the equation as y'-3y=xy². This is known as a Bernoulli
Equation and can be solved using the method given in exercise #37 of Section 2.2.