Systems of firstorder differential equations
189
So long as solutions exist, then the packages will solve the system of equations.
Thus, the system of three equations with initial values:
x
(
t
)
=
x
(
t
)
y
(
t
)
=
x
(
t
)
+
3
y
(
t
)
−
z
(
t
)
z
(
t
)
=
2
y
(
t
)
+
3
x
(
t
)
x
(0)
=
1
,
y
(0)
=
1
,
z
(0)
=
2
(4.36)
can be solved in a similar manner with no difficulty.
In the case of nonlinear systems of differential equations, or where no explicit
solution can be found, then it is possible to use the
NDSolve
command in
Math
ematica
and the
dsolve(. . . , numeric)
command in
Maple
to obtain numerical
approximations to the solutions. These can then be plotted. But often more in
formation can be obtained from direction field diagrams and phase portraits. A
direction field shows a series of small arrows that are tangent vectors to solutions
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 Fall '11
 Dr.Gwartney
 Economics, Differential Equations, Numerical Analysis, Solubility

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