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To summarize, any set of n linearly independent solutions of the system (3) constitutes a fundamental set of solutions. Under the conditions given in this section, such fundamental sets always exist, and every solution of the system (3) can be represented as a linear combination of any fundamental set of solutions.
1. Using matrix algebra, prove the statement following Theorem 7.4.1 for an arbitrary value of the integer k.
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 ofEq. (3) for à < t < â, and let W be the Wronskian of x(1) and x’ (a) Show that
dW _ ~dt =
(b) Using Eq. (3), show that
dx(1) 2() 1 x(1) x(2)
dt dt + dx(V) dx22)
= (P11 + 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.
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 Xj = y and x2 = Ó, then the second order equation
/ + p(t) Ó + q (t) y = 0 (i)
corresponds to the system
Show that if x(j) and x(2) are a fundamental set of solutions of Eqs. (ii), and if y(j) and y(2
X1 = ^
x2 = -q(t)xj - p(t)x2. (ii)
are a fundamental set of solutions of Eq. (i), then W[y(l), y(2)] = cW[x(j), x(2)], where c is a nonzero constant.
Hint: y(j)(t) and y(2)(t) must be linear combinations of xjj(t) and x12(t).
5. Show that the general solution of x' = 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(j)(t) = and x(2)(t) = ^2f)'
(a) Compute the Wronskian of x(j) and x(2).
(b) In what intervals are x(j) and x(2) linearly independent?
(c) What conclusion can be drawn about the coefficients in the system of homogeneous differential equations satisfied by x(j) and x(2)?
(d) Find this system of equations and verify the conclusions of part (c).
7. Consider the vectors x(j) (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(j), ..., x(m) be solutions of x' = P(t)x on the interval a < t < â. 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 < â 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, â) if they are linearly dependent at any point in it.
Hint: There are constants cl,..., cm such that cj x(1) (t0) + ••• + cm x(m) (t0) = 0. Let z(t) = q x(1) ( t) + ••• + cmx(m) (t) and use the uniqueness theorem to show that z ( t) = 0 for each t in a < t < â.
9. Let x(1),..., x(n) be linearly independent solutions of x' = P(t)x, where P is continuous on
a < t < â.
(a) Show that any solution x = z (t) can be written in the form
z(t) = c1x(j)(t) + --- + cn x(n)(t) for suitable constants cj,..., 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(j)(t) + ... + kn x(n)(t), then kj = cv...,kn = cn.
Hint: Show that (kj - cj)x(j)(t) + ••• + (kn - cn)x(n^(t) = 0 for each t in a < t < â and use the linear independence of x(j), ..., 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
— = ax, (2)
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?