520 Ordinary Differential Equations
Using SOLVESYS
The computer program SOLVESYS will attempt to solve numerically the
system initial value problem
(1)Y~ = ii ( x, Y1 , Y2 , · · · , Yn) ;
Y~ = f2(x, Y1, Y2, · · ·, Yn);Y1(c) =di
Y2(c) = d2Y~ = fn(X, Y1, Y2, · · ·, Yn); Yn(c) = dn
on the interval [a, b] for 2 :::; n :::; 6 and c E [a, b]. After the solution has been
calculated you may elect (i) to print the solution components on the monitor,
(ii) to graph any subset of the components in a rectangle R where a :::; x :::; b
and YMIN :::; y:::; YMAX and you select the values for YMIN and YMAX,
or (iii) to produce a phase-plane portrait of Yi versus yj for any two distinct
components Yi and Yj on any rectangle in YjYi-space.
EXAMPLE Solving a System Initial Value Problem
Use SOLVESYS to solve the system initial value problem
dxdt = 3y-2x^2 -1 = f(t,x,y); x(O) = 2
(2)
dy
dt = 8x - y^2 - 7 = g(t, x , y); y(O) = Oon the interval [O, 2.5]. Display a graph of x(t) and y(t). Produce a phase-
plane graph of y(t) versus x(t) on the rectangle
R = { ( x, y) I 0 :::; x :::; 5 and 0 :::; y :::; 5}.
SOLUTION
Notice when using SOLVESYS to solve the system IVP (1), the independent
variable must always be the variable x and the dependent variables must
always be the variables Y1, Y2, ... , Yn· However, in the given system (2) the
independent variable is t and the dependent variables are x and y. When we
specify the system to be integrated to SOLVESYS, n = 2, x will be associated
with Y1, and y will be associated with y2. Making this association, we see that
the system to be solved using the notation of SOLVESYS is
dy1 dx = 3y2 - 2y1 (^2) - 1 = ii ( X, Y1, Y2 ) ; Y1 ( ) 0 = 2
(3)