Download (direct link):
(a) First we show that a and b are linearly independent. Consider the equation c1a + c2b = 0. Express a and b in terms of ^(1) and ^ (1), and then show that (c1 — ic2)?(1) +
(c1 + ic2)!(1) = 0.
(b) Show that c1 — ic2 = 0 and c1 + ic2 = 0 and then that c1 = 0 and c2 = 0. Conse-
quently, a and b are linearly independent.
(c) To show that u(t) and v(t) are linearly independent consider the equation c1u(t0) + c2v(t0) = 0, where t0 is an arbitrary point. Rewrite this equation in terms of a and b, and then proceed as in part (b) to show that c1 = 0 and c2 = 0. Hence u(t) and v(t) are linearly independent at the arbitrary point t0. Therefore they are linearly independent at every point and on every interval.
28. A mass m on a spring with constant k satisfies the differential equation (see Section 3.8)
mu" + ku = 0,
where u(t) is the displacement at time t of the mass from its equilibrium position.
(a) Let x1 = u and x2 = d; show that the resulting system is
x'= ( 0 ') x
(b) Find the eigenvalues of the matrix for the system in part (a).
7.7 Fundamental Matrices
(c) Sketch several trajectories of the system. Choose one of your trajectories and sketch the corresponding graphs of x1 versus t and of x2 versus t. Sketch both graphs on one set of axes.
(d) What is the relation between the eigenvalues of the coefficient matrix and the natural frequency of the spring-mass system?
? 29. Consider the two-mass, three-spring system shown in Figure 7.1.1, whose equations of motion are given in Eqs. (1) of Section 7.1. If there are no external forces, and if the masses and spring constants are equal and of unit magnitude, then the equations of motion are
x7/ = —2x1 + x2, = x1 — 2x2.
(a) Transform the system into a system of four first order equations by letting y1 = x1,
y2 = ^ y3 = xvand y4 = x2.
(b) Find the eigenvalues of the coefficient matrix for the system in part (a).
(c) Solve the system in part (a) subject to the initial conditions yT(0) = (2, 1, 2, 1). Describe the physical motion of the spring-mass system that corresponds to this solution.
(d) Solve the system in part (a) subject to the initial conditions yT(0) = (2, */3, —2, —\/3). Describe the physical motion of the spring-mass system that corresponds to this solution.
(e) Observe that the spring-mass system has two natural modes of oscillation in this case. How are the natural frequencies related to the eigenvalues of the coefficient matrix? Do you think that there might be a third natural mode of oscillation with a different frequency?
7.7 Fundamental Matrices
The structure of the solutions of systems of linear differential equations can be further illuminated by introducing the idea of a fundamental matrix. Suppose that x(1) (t),, x(n) (t) form a fundamental set of solutions for the equation
on some Interval a < t <
x' = P(t)x 1. Then the matrix
whose columns are the vectors x(1) (t),..., x(n) (t), is said to be a fundamental matrix for the system (1). Note that a fundamental matrix is nonsingular since its columns are linearly independent vectors.
Find a fundamental matrix for the system
In Example 1 of Section 7.5 we found that
x(1)(t) = (S) , x(2)(t) = (-2e-t
Chapter 7. Systems ofFirst Order Linear Equations
are linearly independent solutions of Eq. (3). Thus a fundamental matrix for the system (3) is
= (S —2e~t) ? (4)
The solution of an initial value problem can be written very compactly in terms of a fundamental matrix. The general solution of Eq. (1) is
x = c1x(1)(t) +-------+ cn x(n)(t) (5)
or, in terms of V(t),
x = V(t)c, (6)
where c is a constant vector with arbitrary components c1t, cn. For an initial value problem consisting of the differential equation (1) and the initial condition
x(to) = x0, (7)
where t0 is a given point in a < t < j3 and x0 is a given initial vector, it is only necessary to choose the vector c in Eq. (6) so as to satisfy the initial condition (7). Hence c must satisfy
^(tfc = x0. (8)
Therefore, since V(t0) is nonsingular,
c = ^-1(t0)x° (9)
x = V(t)V-1(f0)x0 (10)
is the solution of the initial value problem (1), (7). We emphasize, however, that to solve a given initial value problem one would ordinarily solve Eq. (8) by row reduction and then substitute for c in Eq. (6), rather than compute V-1(t0) and use Eq. (10).
Recall that each column of the fundamental matrix V is a solution of Eq. (1). It follows that V satisfies the matrix differential equation
V = P(t)V. (11)
This relation is readily confirmed by comparing the two sides of Eq. (11) column by column.
Sometimes it is convenient to make use of the special fundamental matrix, denoted by O(f), whose columns are the vectors x(1)(f),..., x(n\t) designated in Theorem
7.4.4. Besides the differential equation (1) these vectors satisfy the initial conditions
7.7 Fundamental Matrices
where e(j) is the unit vector, defined in Theorem 7.4.4, with a one in the jth position and zeros elsewhere. Thus O(t) has the property that