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Ķī+1 = Ķī + 24h(9gn+1 + 19gn - 5gn-1 + gn-2).
Determine an approximate value of the solution at t = 0.4 for the example initial value
problem x = x - 4y, f = -x + y with x(0) = 1, y(0) = 0. Take h = 0.1. Correct the
predicted value once. For the values of x1,..., y3 use the values of the exact solution rounded to six digits: x1 = 1.12883, x2 = 1.32042, x3 = 1.60021, y1 = 0.110527, y2 = -0.250847, and y3 = -0.429696.
REFERENCES There are many books of varying degrees of sophistication dealing with numerical analysis in general and the numerical solution of ordinary differential equations in particular. Among these are:
Ascher, Uri M., and Petzold, Linda R., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (Philadelphia: Society for Industrial and Applied Mathematics, 1998).
Gear, C. William, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice Hall, 1971).
Henrici, Peter, Discrete Variable Methods in Ordinary Differential Equations (New York: Wiley, 1962).
Shampine, Lawrence F., Numerical Solution of Ordinary Differential Equations (New York: Chapman and Hall, 1994).
A detailed exposition of Adams predictor-corrector methods, including practical guidelines for implementation, may be found in:
Shampine, L. F., and Gordon, M. K., Computer Solution of Ordinary Differential Equations: The Initial Value Problem (San Francisco: Freeman, 1975).
Many books on numerical analysis have chapters on differential equations. For example, at an elementary level, see:
Burden, R. L., and Faires, J. D., Numerical Analysis (6th ed.) (Pacific Grove, CA: Brooks/Cole, 1997).
Nonlinear Differential Equations and Stability
There are many differential equations, especially nonlinear ones, that are not susceptible to analytical solution in any reasonably convenient manner. Numerical methods, such as those discussed in the preceding chapter, provide one means of dealing with these equations. Another approach, presented in this chapter, is geometrical in character and leads to a qualitative understanding of the behavior of solutions rather than detailed quantitative information.
9.1 The Phase Plane: Linear Systems
ŠSince many differential equations cannot be solved conveniently by analytical methods, it is important to consider what qualitative1 information can be obtained about their solutions without actually solving the equations. The questions that we consider in this chapter are associated with the idea of stability of a solution, and the methods that we employ are basically geometrical. Both the concept of stability and the use of
1The qualitative theory of differential equations was created by Henri Poincare (1854-1912) in several major papers between 1880 and 1886. Poincare was professor at the University of Paris and is generally considered the leading mathematician of his time. He made fundamental discoveries in several different areas of mathematics, including complex function theory, partial differential equations, and celestial mechanics. In a series of papers beginning in 1894 he initiated the use of modern methods in topology. In differential equations he was a pioneer in the use of asymptotic series, one of the most powerful tools of contemporary applied mathematics. Among other things, he used asymptotic expansions to obtain solutions about irregular singular points, thereby extending the work of Fuchs and Frobenius discussed in Chapter 5.
Chapter 9. Nonlinear Differential Equations and Stability
geometric analysis were introduced in Chapter 1 and used in Section 2.5 for first order autonomous equations
dy/dt = f (y). (1)
In this chapter we refine the ideas and extend the discussion to systems of equations.
We start with a consideration of the simplest system, namely, a second order linear homogeneous system with constant coefficients. Such a system has the form
dx/dt = Ax, (2)
where A is a 2 x 2 constant matrix and x is a 2 x 1 vector. Systems of this kind were solved in Sections 7.5 through 7.8. Recall that if we seek solutions of the form x = gert, then by substitution for x in Eq. (2) we find that
(A - rI)g = 0. (3)
Thus r must be an eigenvalue and g a corresponding eigenvector of the coefficient matrix A. The eigenvalues are the roots of the polynomial equation