ABSTRACT
In Chapter 1, the solution of a first-order differential equation was considered. Higher derivatives can occur in some problems, and so we are led to the general ordinary differential equation
F
( t, y,
dy
dt , d2y
dt2 , . . . ,
dny
dtn
) = 0, (2.1.1)
where F is some function with n + 2 arguments. It is called ordinary because it involves only the ordinary derivatives of y with respect to the single variable t. Later on we will study cases in which y is a function of more than one variable. In that case, partial derivatives of y can arise and a partial differential equation has to be solved. The order of an ordinary differential equation is the order of the highest
derivative appearing. Thus
d4y
dt4 =
dy
dt
( d3y
dt3
)4 + y2
is of order 4, whereas ( dy
dt
)2 = t2 + y2
is of order 1. The two main categories into which ordinary differential equations are clas-
sified are linear and nonlinear. The form of the general linear ordinary differential equation of order n is
an(t) dny
dtn + an−1(t)
dn−1y dtn−1
+ · · ·+ a1(t)dy dt
+ a0(t)y = f(t), (2.1.2)
where a0(t), . . . , an(t) are known functions of t. If all of a0(t), . . . , an(t) are constants, (2.1.2) is known as a linear ordinary differential equation with
does have the structure of (2.1.2) is called nonlinear; it will contain products such as
y2, dy
dt
d2y
dt2
or functions such as ey. For example,
t d2y
dt2 + et
dy
dt + y cos t = t3 tan t
is linear and of order 2,
5 d2y
dt2 + 4
dy
dt + 3y = ln2 t
is linear with constant coefficients and of order 2, while
y2 dy
dt = t
is nonlinear. The solution of an ordinary differential equation is always sought on an
interval (a, b) (a < b) of t. It is a relation between y and t which satisfies the ordinary differential equation when t is any point of the interval and does not contain any derivatives or integrals of y. Integrals of functions of t may be involved but these should be evaluated when it is reasonable to do so. The general solution (sometimes called the complete primitive) of an
ordinary differential equation of order n must contain n arbitrary constants. Any solution that does not have n arbitrary constants is not the general solution. For instance, you can check that
y = et − 1/t (2.1.3) satisfies the ordinary differential equation
t3 ( d2y
dt2 − y
) = t2 − 2, (2.1.4)
but it is not the general solution because the general solution must contain two arbitrary constants whereas (2.1.3) has none. Similarly
y = Ce−t − 1/t, (2.1.5) with C an arbitrary constant, is a solution of (2.1.4) but is not the general solution. On the other hand,
y = C1e t + C2e
−t − 1/t, (2.1.6) with C1 and C2 arbitrary constants, is the general solution of (2.1.4). When additional information is available, it may be possible to assign par-
ticular values to the arbitrary constants in the general solution. For example,
such that y dy/dt when t = 1. From the general solution (2.1.6)
dy/dt = C1e t − C2e−t + 1/t2,
and so the conditions at t = 1 can be satisfied if
C1e+ C2e −1 − 1 = 1,
C1e− C2e−1 + 1 = 1. These require that C1 = e
−1 and C2 = e; hence
y = et−1 + e1−t − 1/t is the solution of (2.1.4) which takes the correct values at t = 1. An ordinary differential equation may not possess a solution. There are
theorems, called existence theorems, which tell you that certain types of differential equations have a solution. If a solution exists and you can prove that it is the only one that satisfied any conditions imposed, then you have demonstrated a uniqueness theorem. Existence and uniqueness are beyond the scope of this chapter (for some information, see Section 5.11). Even when existence and uniqueness theory is available, the actual finding of
a solution may be a difficult task. For instance, there is an existence theorem for linear ordinary differential equations with variable coefficients, but the solution cannot always be written down easily. Again, the solution of
dy
dt = t2 + y2
cannot be expressed in terms of elementary functions although the solution is known to exist. Therefore, one must turn to numerical techniques, as discussed for example in Section 1.7, to find an approximation to the solution. This chapter will be confined to discussing linear ordinary differential equa-
tions with constant coefficients. For these, not only is existence theory available but also the general solution can be determined explicitly.
