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is a fundamental matrix. Then the solution x of Eq. (35) is given by x = ^(')u('), where u(') satisfies ^(')u'(') = g('), or
e-3' e"'\ /VA (2e
— e"3t e"^ W V 3t Solving Eq. (37) by row reduction, we obtain
u[ = e2t — 3 te3t,
u'2 = 1 + 3 t?.
u 1(t) = Ie2t - 1 t?3t + 6?3t + q, u2(t) = t + 2 tet — 33 et + c2,
x = ^(t)u(t)
=c, (-!)e-3t+(!)e-t+te~'+1 (-^e-t + (0'- 3 (0 • (38)
which is the same as the solution obtained previously.
Each of the methods for solving nonhomogeneous equations has some advantages and disadvantages. The method of undetermined coefficients requires no integration, but is limited in scope and may entail the solution of several sets of algebraic equations. The method of diagonalization requires finding the inverse of the transformation matrix and the solution of a set of uncoupled first order linear equations, followed by a matrix multiplication. Its main advantage is that for Hermitian coefficient matrices the inverse of the transformation matrix can be written down without calculation, a feature that is more important for large systems. Variation of parameters is the most general method. On the other hand, it involves the solution of a set of linear algebraic equations with variable coefficients, followed by an integration and a matrix multiplication, so it may also be the most complicated from a computational viewpoint. For many small systems
7.9 Nonhomogeneous Linear Systems
with constant coefficients, such as the one in the examples in this section, there may be little reason to select one of these methods over another. Keep in mind, however, that the method of diagonalization is slightly more complicated if the coefficient matrix is not diagonalizable, but only reducible to a Jordan form, and the method of undetermined coefficients is practical only for the kinds of nonhomogeneous terms mentioned earlier.
For initial value problems for linear systems with constant coefficients, the Laplace transform is often an effective tool also. Since it is used in essentially the same way as described in Chapter 6 for single scalar equations, we do not give any details here.
In each of Problems 1 through 12 find the general solution of the given system of equations.
3x'= 2 -2 x + - COS :
5. «? = F hx +, :-2
2- x = C? -1 |x+
4. « = 14 -2) X + ( f 2e'
-2 t '
t > 0
6 x'=( 2 -1)x + (2t--i+4
7 x'=(4 0x+(-et
9. x' =
t > 0
8 •'=(3 -?)xH-^et
10. •'' = (-3 -2)•+(-0^
11. • = 11 2 ) • +1 t
,1 —2 \cos t
12."= 1 -5 • + cec ;
0 < t < n
13. The electric circuit shown inFigure 7.9.1 is described by the system of differential equations
I (t), (i)
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.
16. Let x = ^(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 ^(t) _ v(t), show that ^(t) = 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 = ®(t)x0 + f &(t _ s)g(s) ds.
(b) Show also that
x = exp(At)x0 + f exp[A(t _ s)]g(s) ds.
Compare these results with those ofProblem 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.