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Chapter 4. Higher Order Linear Equations
simply state the results for the nth order problem. The proofs of most of the results are also similar to those for the second order equation and are usually left as exercises.
Since Eq. (2) involves the nth derivative of y with respect to t, it will, so to speak, require n integrations to solve Eq. (2). Each of these integrations introduces an arbitrary constant. Hence we can expect that, to obtain a unique solution, it is necessary to specify n initial conditions,
y(t0) = yo> y (to) = y0 > ???> y(n-1)(to) = y0n-1)> (3)
where t0 may be any point in the interval I and y0, y0,, y0n- ^ is any set of prescribed real constants. That there does exist such a solution and that it is unique are assured by the following existence and uniqueness theorem, which is similar to Theorem 3.2.1.
If the functions p1, p2,..., pn, and g are continuous on the open interval I, then there exists exactly one solution y = fi(t) of the differential equation (2) that also satisfies the initial conditions (3). This solution exists throughout the interval I.
We will not give a proof of this theorem here. However, if the coefficients p17..., pn are constants, then we can construct the solution of the initial value problem (2), (3) much as in Chapter 3; see Sections 4.2 through 4.4. Even though we may find a solution in this case, we do not know that it is unique without the use of Theorem 4.1.1. A proof of the theorem can be found in Ince (Section 3.32) or Coddington (Chapter 6).
The Homogeneous Equation. As in the corresponding second order problem, we first discuss the homogeneous equation
L [y] = y(n) + p, (t) y(n-1) + + pn-l(t)y + pn (t) y = 0. (4)
If the functions y1, y2,..., yn are solutions of Eq. (4), then it follows by direct computation that the linear combination
y = c1 y1(t) + c2 y2(t) + + cyn (t)> (5)
where c17..., cn are arbitrary constants, is also a solution of Eq. (4). It is then natural to ask whether every solution of Eq. (4) can be expressed as a linear combination of yv ..., yn. This will be true if, regardless of the initial conditions (3) that are prescribed, it is possible to choose the constants cv ..., cn so that the linear combination (5)
satisfies the initial conditions. Specifically, for any choice of the point t0 in I, and for
any choice of y0, y0,..., y0n-1), we must be able to determine cv ..., cn so that the equations
c1 y1(t0) + " + cnyn (t0) = y0
c1 y1 (t0) + -'- + cJn (t0) = y0 ...
c1 y[n-1)(t0) + + cnyin-l)(t0) = y0n-1)
are satisfied. Equations (6) can be solved uniquely for the constants cv ..., cn, provided that the determinant of coefficients is not zero. On the other hand, if the determinant of coefficients is zero, then it is always possible to choose values of y0, y0,..., y0n-1)
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 y1, 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 of Eq. (4) can be expressed as a linear combination of the solutions y1, y2,..., yn.
A set of solutions y1,..., 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 of Eq. (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 of Eq. (4).
The discussion of linear dependence and independence given in Section 3.3 can also be generalized. The functions f1, 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 f1t..., fn are said to be linearly independent on I if they are not linearly dependent there. If y1,..., 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 of Eq. (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