# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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2. In this problem we outline a proof of Theorem 7.4.3 in the case n = 2. Let x(1) and x(2) be

solutions of Eq. (3) for a < t < j3, and let Wbe the Wronskian of x-1- and (a) Show that

dW _ ~dt ~

(b) Using Eq. (3), show that

,(2)

dx(1) dx(2 x((1) x(2)

dt dt x (2) x2 + dx21) dx22)

dt dt

dW

= (Pu + P22) W'

(c) Find W(t) by solving the differential equation obtained in part (b) . Use this expression to obtain the conclusion stated in Theorem 7.4.3.

372

Chapter 7. Systems ofFirst Order Linear Equations

(d) Generalize this procedure so as to prove Theorem 7.4.3 for an arbitrary value of n.

3. Show that the Wronskians of two fundamental sets of solutions of the system (3) can differ at most by a multiplicative constant.

Hint: Use Eq. (14).

4. If x1 = y and x2 = ?, then the second order equation

corresponds to the system

/ + p(t) y + q (t) y = 0 ()

*1 = ^

x2 = -q(t)x1 - p(t)x2. (ii)

Show that if x(1) and x(2) are a fundamental set of solutions of Eqs. (ii), and if y(1) and y(2) are a fundamental set of solutions of Eq. (i), then W[y(l), y(2)] = cW[x(1), x(2)], where c is a nonzero constant.

Hint: y(V)(t) and y(2)(t) must be linear combinations of x11(t) and x12(t).

5. Show that the general solution of x1 = P(t)x + g(t) is the sum of any particular solution x(p) of this equation and the general solution x(c) of the corresponding homogeneous equation.

6. Consider the vectors x(1)(t) = and x(2)(t) = ^2f)'

(a) Compute the Wronskian of x(1) and x(2).

(b) In what intervals are x(1) and x(2) linearly independent?

(c) What conclusion can be drawn about the coefficients in the system of homogeneous differential equations satisfied by x(1) and x(2)?

(d) Find this system of equations and verify the conclusions of part (c).

7. Consider the vectors x(1) (t) = ^2t^ and x(2) (t) = , and answer the same questions as

in Problem 6.

The following two problems indicate an alternative derivation of Theorem 7.4.2.

8. Let x(1), ..., x(m) be solutions of x1 = P(t)x on the interval a < t < p. Assume that P is continuous and let t0 be an arbitrary point in the given interval. Show that x(1), ..., x(m) are linearly dependent for a < t < p if (and only if) x(1) (t0), ..., x(m) (t0) are linearly dependent. In other words x(1),..., x(m) are linearly dependent on the interval (a, p) if they are linearly dependent at any point in it.

Hint: There are constants c1,..., cm such that c1x(1)(t0) + ••• + cmx(m)(t0) = 0. Letz(t) = c1x(1) ( t) + ••• + cmx(m) (t) and use the uniqueness theorem to show that z( t) = 0 for each t in a < t < p.

9. Let x(1),..., x(n) be linearly independent solutions of x1 = P(t)x, where P is continuous on

a < t < p.

(a) Show that any solution x = z(t) can be written in the form

z(t) = c1x(1)(t) +-------+ cnx(n\t)

for suitable constants c1,..., cn.

Hint: Use the result of Problem 11 of Section 7.3, and also Problem 8 above.

(b) Show that the expression for the solution z(t) in part (a) is unique; that is, if z(t) = k1x(1) (t) + ••• + kn x(n) (t), then k1 = cv...,kn = cn.

Hint: Show that (k1 - c1)x(1)(t) + ••• + (kn - cn)x(n')(t) = 0 for each t in a < t < p and use the linear independence of x(1),..., x(n).

7.5 Homogeneous Linear Systems with Constant Coefficients

373

7.5 Homogeneous Linear Systems with Constant Coefficients

We will concentrate most of our attention on systems of homogeneous linear equations with constant coefficients; that is, systems of the form

x = Ax, (1)

where A is a constant n x n matrix. Unless stated otherwise, we will assume further that all the elements of A are real (rather than complex) numbers.

If n = 1, then the system reduces to a single first order equation

dx

— = ax, (2)

dt W

whose solution is x = ceat. In Section 2.5 we noted that x = 0 is the only equilibrium solution if a = 0. Other solutions approach x = 0 if a < 0 and in this case we say that x = 0 is an asymptotically stable equilibrium solution. On the other hand, if a > 0, then x = 0 is unstable, since other solutions depart from it. For higher order systems the situation is somewhat analogous, but more complicated. Equilibrium solutions are found by solving Ax = 0. We assume that det A = 0, so x = 0 is the only equilibrium solution. An important question is whether other solutions approach this equilibrium solution or depart from it as t increases; in other words, is x = 0 asymptotically stable or unstable? Or are there still other possibilities?

The case n = 2 is particularly important and lends itself to visualization in the x1 x2-plane, called the phase plane. By evaluating Ax at a large number of points and plotting the resulting vectors one obtains a direction field of tangent vectors to solutions of the system of differential equations. A qualitative understanding of the behavior of solutions can usually be gained from a direction field. More precise information results from including in the plot some solution curves, or trajectories. A plot that shows a representative sample of trajectories for a given system is called a phase portrait. Examples of direction fields and phase portraits occur later in this section.

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