112 Ordinary Differential Equations
rectangle which does not contain a point of the x-axis, where y = 0. So by the
fundamental theorems there exists a unique solution of the IVP (28) and this
solution can be continued in a unique manner until either x -> -oo, x -> +oo,
y , 0 (since existence and uniqueness may no longer be guaranteed at y = 0) ,
or y , +oo (since the y coordinate of the initial condition, y(-1) = 1 , is
positive).
Numerical Solution
A graph of the numerical solution to the IVP (28) generated using MAPLE
is shown in Figure 2.16. The following four MAPLE statements were used to
produce the graph.
with(DEtools) :with(plots):de:=diff(y(x), x)= -x/y(x):
p:=DEplot( de,y(x),x=-2 .. 2,{ [y(-1 )=1 J },arrows=NONE,axes=BOXED,view=[-2.2 .. 2.2,-2.2 .. 2.2]):display(p);The second statement specifies the differential equation of (28). The third
statement specifies the interval of integration with x = -2 .. 2 and the
initial condition with y( -1) = -1. We included the specification: view=
[-2.2 .. 2.2, -2.2 .. 2.2] because we wanted the horizontal axis of the graph to
be slightly longer than the interval of integration and because we wanted to
control the range of the y values instead of letting the software do so. The
solution is "reasonably" accurate-but, of course, not exact- on the inter-
val [-1.4, 1.4] where the computed solution is positive. The solution of the
IVP (28) is y(x) = )2 - x^2. (Verify this fact.) This solution exists only on
the interval (-v'2, v'2). The graph of the solution is the upper half of a circle
with center at the origin and radius v'2.
The general solution of the differential equation of the IVP (28), y' = -x/y,
is x^2 + y^2 = k^2 , where k is an arbitrary constant. For k f. 0 the graph of
x^2 + y^2 = k^2 is a circle with center at the origin and radius jkj. Thus, the
graph of the one-parameter family of curves x^2 + y^2 = k^2 is the set of all
concentric circles with center at the origin. Look at Figure 2.16 again. Notice
for -2 < x < -1.4 and for 1.4 < x < 2 the computer generated solution
is attempting to approximate other members of the general solution of the
differential equation.