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4.1 General Theory of nth Order Linear Equations
such that Eqs. (6) do not have a solution. Hence a necessary and sufficient condition
is not zero at t = t0. Since t0 can be any point in the interval I, it is necessary and sufficient that W(yx,y2,..., yn) be nonzero at every point in the interval. Just as for the second order linear equation, it can be shown that if yv y2,, yn are solutions of Eq. (4), then W(y1, y2,..., yn) is either zero for every t in the interval I or else is never zero there; see Problem 20. Hence we have the following theorem.
Theorem 4.1.2 If the functions px, p2,..., pn are continuous on the open interval I, if the functions
y1, y2,..., yn are solutions of Eq. (4), and if W(y1; y2,..., yn)(t) = 0 for at least one point in I, then every solution ofEq. (4) can be expressed as a linear combination of the solutions ë y2,..., yn.
A set of solutions yl,..., yn of Eq. (4) whose Wronskian is nonzero is referred to as a fundamental set of solutions. The existence of a fundamental set of solutions can be demonstrated in precisely the same way as for the second order linear equation (see Theorem 3.2.5). Since all solutions ofEq. (4) are of the form (5), we use the term general solution to refer to an arbitrary linear combination of any fundamental set of solutions ofEq. (4).
The discussion of linear dependence and independence given in Section 3.3 can also be generalized. The functions fl, f2,..., fn are said to be linearly dependent on I if there exists a set of constants kv k2,..., kn, not all zero, such that
for all t in I. The functions fv ..., fn are said to be linearly independent on I if they are not linearly dependent there. If yv ..., yn are solutions of Eq. (4), then it can be shown that a necessary and sufficient condition for them to be linearly independent is that W(y1;..., yn)(t0) = 0 for some t0 in I (see Problem 25). Hence a fundamental set of solutions of Eq. (4) is linearly independent, and a linearly independent set of n solutions ofEq. (4) forms a fundamental set of solutions.
The Nonhomogeneous Equation. Now consider the nonhomogeneous equation (2),
If Y1 and Y2 are any two solutions of Eq. (2), then it follows immediately from the linearity of the operator L that
k1 f1 + k2 f2 + + knfn = 0
L[y] = y(n) + p, (t)y(n-l) + ••• + pn(t)y = g(t).
LY - Y2KO = L[YJW - L[Y2KO = g(t) - g(t) = 0.
Hence the difference of any two solutions of the nonhomogeneous equation (2) is a solution of the homogeneous equation (4). Since any solution of the homogeneous
Chapter 4. Higher Order Linear Equations
equation can be expressed as a linear combination of a fundamental set of solutions yv , yn, it follows that any solution of Eq. (2) can be written as
y = cx ó() + c2y2(t) + ••• + cnyn (t) + Y (t), (9)
where Y is some particular solution of the nonhomogeneous equation (2). The linear combination (9) is called the general solution of the nonhomogeneous equation (2).
Thus the primary problem is to determine a fundamental set of solutions yl,..., yn of the homogeneous equation (4). If the coefficients are constants, this is a fairly simple problem; it is discussed in the next section. If the coefficients are not constants, it is usually necessary to use numerical methods such as those in Chapter 8 or series methods similar to those in Chapter 5. These tend to become more cumbersome as the order of the equation increases.
The method of reduction of order (Section 3.5) also applies to nth order linear equations. If y is one solution of Eq. (4), then the substitution y = v(t)y1(t) leads to a linear differential equation of order n - 1 for v' (see Problem 26 for the case when n = 3). However, if n > 3, the reduced equation is itself at least of second order, and only rarely will it be significantly simpler than the original equation. Thus, in practice, reduction of order is seldom useful for equations of higher than second order.
In each of Problems 1 through 6 determine intervals in which solutions are sure to exist.
1. yv + 4Ó + 3y = t 2. ty" + (sin t)y" + 3y = cos t
3. t(t - 1)yv + eff + 4t2y = 0 4. Ó + ty + t2Ó+ t3y = ln t
5. (x - 1)yv + (x + 1)y" + (tanx)y = 0 6. (x2 - 4)yvi + x2Ó + 9y = 0
In each of Problems 7 through 10 determine whether the given set of functions is linearly dependent or linearly independent. If they are linearly dependent, find a linear relation among them.
7. f1(t) = 2t - 3, f2(t) = t2 + 1, f3(t) = 2t2 - t
8. f() = 2t - 3, f2(t) = 2t2 + 1, f3(t) = 3t2 + t
9. f1(t) = 2t - 3, f2(t) = t2 + 1, f3(t) = 2t2 - t, f4(t) = t2 + t + 1