### Dr. Sarah's Comments and Peer Review for James and Jeremy's 10

Change the dfieldplot command to increase the values of y, sinc there is interesting behavior occuring near 4 and 5. dfieldplot(diff(y(x),x)=3*sin(y(x))+y(x)-2,y(x),x=-2..3,y=-1..6,arrows=LINE,axes=BOXED);

fsolve(3*sin(y)+y-2=0,y,0..1); .5170489637 this is one equilibrium solution, and we can see from the dfieldplot that this is an unstable solution since other solutions above and below it tend away from x=.517....

fsolve(3*sin(y)+y-2=0,y,3..4); 3.774518012 is a stable solution

fsolve(3*sin(y)+y-2=0,y,4..5); 4.929543438 is an unstable solution

2nd part - good, but let me clarify the y=x curve a bit. Notice that y=x (as a curve) is NOT a solution of the de. Let's try plugging it in to the d.e. to see this: Well, if y=x as a curve was a solution, then dy/dx=dx/dx =1. So, the lhs of the de is 1. But the rhs of the de is y^2-yx = x^2-x^2 when substituting y=x, so we get a 0 on the rhs of the de. Since 1 is not equal to 0, y=x is not a solution as a curve of the de.

Let's investigate what happens to a solution with an initial condition that lies on y=x. (Say y(4)=4, or y(20)=20). Our initial point (x,x) is on the dfieldplot, and we know that the solution will not stay along the curve y=x, so let's see where it goes. Now we know that initially, the derivative at the point (x,x) will be 0 via the rhs of the de. Hence, the solution curve will travel horizontally a bit to the right (we can see this on the dfieldplot). Well, once this has occurred, then we are below the y=x curve and so our solutions x value is now larger than our solution's y value. It then follows the behavior of the rest of the solutions that are below the y=x curve - it tends towards 0 because the rhs of the de (y^2 -xy) is negative since x>y. Then, the solution curve ends up decreasing.

### Peer Review

good
good job on catching their mistakes. Good job on a very confusing problem
They seemed to have a very tough problem and did a fairly good job with it.
worksheet didn't clearly show what was going on (re-state the problems) but the work looks good and they explained it well.
1st plot needs larger x and y ranges! Interesting activity above y=4, nice plots, good description of steps
a few things confusing, but looks good
good presentation although kind of unorganized.
dfieldplot is very useful, good answers to Dr. Sarah's questions
good maple work, good talking
graphs look good, what is the equation at the end? Didn't understand the explanation of this equation.
What is the DE or where is the DE? Jumps right into the dfield, good job.
Maple presentation lacks the original diff eqns, so I got a little confused when it came to the second part.
good, but weren't real sure on their explanation.
They obviously tried hard, but seemed kind of confused on some of the concepts.
Worksheet a little confusing - need to related problems and parts somewhere in the worksheet. It was hard to tell what related to what part. Otherwise, good work.
Hard to follow at start. Need to clarify. Didn't answer all ?'s on Maple.
You might want to remember to include your de's. Might want to explain what plots mean. Maple syntax could be cleared up some. You might explain your syntax a bit more.
They explained their mistakes, it was good!
Slowed down at end and that helped immensely.