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where x1 is the current through the inductor, x2 is the voltage drop across the capacitor, and I(t) is the current supplied by the external source.
(a) Determine a fundamental matrix ^(f) for the homogeneous system corresponding to Eq. (i). Refer to Problem 25 of Section 7.6.
L = 8 henrys
FIGURE 7.9.1 The circuit in Problem 13.
Chapter 7. Systems of First Order Linear Equations
(b) If I(t) = e t/2, determine the solution of the system (i) that also satisfies the initial conditions x(0) = 0.
In each of Problems 14 and 15 verify that the given vector is the general solution of the corresponding homogeneous system, and then solve the nonhomogeneous system. Assume that t > 0.
14. tx' = (2 -1¿ x + Ï - ^ xM = - Ï\. + Ï
3 -2 2t 1 1 2 3
2 -2)x+C-^), x(c) = ^ (2)t-1+^t2
16. Let x = c^(t) be the general solution of x' = P(t)x + g(t), and let x = v(t) be some particular solution of the same system. By considering the difference ô(0 - v(t), show that ô(0 = u(t) + v(t), where u(t) is the general solution of the homogeneous system x' = P(t)x.
17. Consider the initial value problem
x'= Ax + g(t), x(0) = x0.
(a) By referring to Problem 15(c) in Section 7.7, show that
x = Ô^0 + f Ô(t - s)g(s) ds.
(b) Show also that
x = exp(At)x0 + I exp[A(t - s)]g(s) ds.
Compare these results with those of Problem 27 in Section 3.7.
Further information on matrices and linear algebra is available in any introductory book on the subject. The following is a representative sample:
Anton, H., and Rorres, C., Elementary Linear Algebra (8th ed.) (New York: Wiley, 2000).
Johnson, L. W., Riess, R. D., and Arnold, J. T., Introduction to Linear Algebra (4th ed.) (Reading, MA: Addison-Wesley, 1997).
Kolman, B., Elementary Linear Algebra (7th ed.) (Upper Saddle River, NJ: Prentice Hall, 1999). Leon, S. J., Linear Algebra with Applications (4th ed.) (New York: Macmillan, 1994).
Strang, G., Linear Algebra and Its Applications (3rd ed.) (New York: Academic Press, 1988).
Up to this point we have discussed methods for solving differential equations by using analytical techniques such as integration or series expansions. Usually, the emphasis was on finding an exact expression for the solution. Unfortunately, there are many important problems in engineering and science, especially nonlinear ones, to which these methods either do not apply or are very complicated to use. In this chapter we discuss an alternative approach, the use of numerical approximation methods to obtain an accurate approximation to the solution of an initial value problem. The procedures described here can be executed easily on personal computers as well as on some pocket calculators.
8.1 The Euler or Tangent Line Method
To discuss the development and use of numerical procedures we will concentrate mainly on the first order initial value problem consisting of the differential equation
² = f(t ¦y' (1)
and the initial condition
y(to) = Óî- (2)
We assume that the functions f and f are continuous on some rectangle in the ty-plane containing the point (t0, y0). Then, by Theorem 2.4.2, there exists a unique solution
Chapter 8. Numerical Methods
y = <p( t) of the given problem in some interval about t0. If Eq. (1) is nonlinear, then the interval of existence of the solution may be difficult to determine and may have no simple relationship to the function f. However, in all our discussions we assume that there is a unique solution of the initial value problem (1), (2) in the interval of interest.
In Section 2.7 we described the oldest and simplest numerical approximation method, namely, the Euler or tangent line method. This method is expressed by the equation
Óï+1 = Óï + f(tn, óï)(tn+1 - tn), n = 0, 1, 2,.... (3)
If the step size has a uniform value h and if we denote f (tn, yn) by fn, then Eq. (3) simplifies to
Óï+1 = 7n + fnh, n = 0 1 2>---- (4)
Euler’s method consists of repeatedly evaluating Eq. (3) or (4), using the result of each step to execute the next step. In this way you obtain a sequence of values y0, yv y2,, yn,... that approximate the values of the solution ô() at the points