Systems of first-order differential equations189So 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)+3y(t)−z(t)z(t)=2y(t)+3x(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 explicitsolution can be found, then it is possible to use theNDSolvecommand inMath-ematicaand thedsolve(. . . , numeric)command inMapleto obtain numericalapproximations to the solutions. These can then be plotted. But often more in-formation can be obtained from direction field diagrams and phase portraits. Adirection field shows a series of small arrows that are tangent vectors to solutions
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