1550078481-Ordinary_Differential_Equations__Roberts_

114 Ordinary Differential Equations

we see that a solution exists as long as (x, y(x)) remains in the first or third
quadrant or y(x) = 0. Furthermore, the solution is unique as long as (x, y(x))
remains in the first or third quadrant. Once the solution, y(x), reaches the
x-axis- that is , once y(x) = 0- the solution may no longer be unique.

a. The point ( -1, 0), which corresponds to the initial condition y( -1) = 0,

is on the x-axis so the solution to the IVP (29a) exists, but the solution

may not be unique. Notice that y(x) = 0 is a solution of (29a). Verify

that for all r:::; -1 and alls::'.:'. 0

-[(-x)^312 - r]2 /9, r:::; -1

Yrs(x) = 0 -1 < x < 0

(x^312 -s)^2 /9 o:::;s

is a solution of the IVP (29a). Hence (29a) has an infinite number of

so lutions on ( -oo, oo).

b. Since (-1, -1 /9) is in the third quadrant, the IVP (29b) has a unique

solution until x , -oo, x , o-, y , -oo, or y , o-. On the interval

(-oo,O) the unique solution to the IVP (29b) is y(x) = x^3 /9. (Verify

this fact.) At x = 0, y(x) = 0 and the so lution may no longer be unique.

Show that for all u ::'.:'. 0

l

x3/9

Yu(x) = 0 :

(x3/2 - u)2 /9 '

O<x<u

u:::;x

is a solution of the IVP (29b). Hence, (29b) has an infinite number of

solutions on (-oo, oo ).

Numerical Solution

We generated numerical approximations to the IVPs (29a) and (29b) using
both MAPLE and SOLVEIVP.

a. The initial point (-1, 0) is on the x-axis and as we saw in the math-

ematical analysis section, there are an infinite number of solutions to

the IVP (29a). SOLVEIVP generated the so lution y(x) = 0. The soft-

ware gave us no indication that this is not a unique solution. MAPLE

indicated floating point overflow no matter how small the interval of

integration was chosen. This result would often lead one to assume the

IVP (29a) has no solution.