Applications of Systems of Equations 437SOLUTIONMathematical Analysis The functions fx = -4x, fy = 3, 9x = 8, and
gy = -2y are all defined and continuous on the entire xy-plane. Hence,
according to the existence and uniqueness theorem and the continuation the-
orem presented in chapter 7, the system initial value problem (8) has a unique
solution on some interval I centered about c = 0 and this solution can be ex-
tended uniquely until either x(t) ---> ±oo or y(t) ---> ±oo. Since the functions
x(t) and y(t) have physical significance only for x(t) 2 0 and y(t) ;::: 0, we will
stop the numerical integration if x(t) or y(t) becomes negative.
Computer Solution We input the two functions defining the system,
f(x, y) and g(x, y); the interval of integration [O, 2.5]; and the initial conditionsx(O) = 2 and y(O) = 0 into our computer software. After the integration was
completed, we indicated we wanted to graph x(t) and y(t) on the rectangle
R = {(t, x) I 0:::::; t:::::; 2.5 and 0:::::; x:::::; 5}. The resulting graph is displayed in
Figure 10.10. The graph of x(t) starts at a height of 2, decreases to nearly 1,
and then increases to 1.87154. The graph of y(t) starts at a height of 0 and
steadily increases to 2.743162.
Figure 10.10 Graph of the Solution of System (8)Since we wanted to produce a phase-plane graph of y versus x, we indicated
to our computer software SOLVESYS that we wanted x(t) assigned to the
horizontal axis and y(t) assigned to the vertical axis and that we wanted to