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An elementary book on electrical circuits is:
Bobrow, L. S., Elementary Linear Circuit Analysis (New York: Oxford University Press, 1996).
Higher Order Linear Equations
The theoretical structure and methods of solution developed in the preceding chapter for second order linear equations extend directly to linear equations of third and higher order. In this chapter we briefly review this generalization, taking particular note of those instances where new phenomena may appear, due to the greater variety of situations that can occur for equations of higher order.
4.1 General Theory of nth Order Linear Equations
An nth order linear differential equation is an equation of the form
dn y dn-1 y dy
P0(t) dy + Px(t) -yy^ + + Pn-1(t) dy + pn (t )y = G(t). (1)
We assume that the functions P0,, Pn and G are continuous real-valued functions on some interval I: a < t < â, and that P0 is nowhere zero in this interval. Then, dividing Eq. (1) by P0(t), we obtain
dny dn-1 y dy
L [y] = dn+p1(t) dt + ¦¦¦+pn-1(t) dt + pn(t) y=g(t) (2)
The linear differential operator L of order n defined by Eq. (2) is similar to the second order operator introduced in Chapter 3. The mathematical theory associated with Eq. (2) is completely analogous to that for the second order linear equation; for this reason we
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) = y0 y (t0) = ó0 > > y(n-1)(t0) = ó0Ë-1)> (3)
where t0 may be any point in the interval I and y0, y0, , y0n-1) 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 = ô^) 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-,(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 c1t , c 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 y1, , yn. This will be true if, regardless of the initial conditions (3) that are prescribed, it is possible to choose the constants c1t , c 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, , ó0-1), we must be able to determine c1t , cn so that the equations
c1 y1(t0) + ¦¦¦ + cnyn (t0) = y0
c1 y1 (t0) + -^ + cnyn (t0) = y0 _
c1 yr1](t0) + ¦¦¦ + cnynn-1)(t0) = y0n-1)
are satisfied. Equations (6) can be solved uniquely for the constants c1t , 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, , ó0-1)