42 Ordinary Differential Equations
In exercises 6-17 graph the direction field of the given differential
equation on the rectangle
R={(x,y)f -5 :S: x :S: 5 and -5 :S: y :S: 5}.
When possible, indicate where the direction field is undefined, where
solutions are increasing and decreasing, where relative maxima and
relative minima occur, and the asymptotic behavior of solutions.
- y'=x+y 7. y' = xy
8. y' = x/y 9. y' = y/x
y' = 1 + y2 11. y' = y2 - 3y
y' = x3 + y3 13. y' = IYI
y' = ex-y 15. y' = ln(x + y)
16.2x-y
17.
I 1
y'=-- y =x+3y ./15 - x^2 -y^2
1 8. Graph the direction field for y' = 3y^213 on the rectangleR = {(x, y) I - 5 :S: x :S: 5 and - 5 :S: y :S: 5}.
(HINT: Enter y^213 as yA(2/3) and as (yA2t(l/3). Notice the difference in thegraphs. Which graph is the correct direction field for y' = 3y^213 ?)
2. 2 Fundamental Theorems
By stating and discussing three fundamental theorems regarding the initial
value problem(1) y' = j(x, y); y(c) = d
we hope to answer, at least in part, the following three questions:
"Under what conditions does a solution to the IVP (1) exist?"
"Under what conditions is the solution to the IVP (1) unique?"
"Where- that is, on what interval or what region- does the solution to the
IVP (1) exist and where is the solut ion unique?"