1. Standard Form
A second order linear differential equation is a differential equation of the form
(Here A,B,C and D are certain prescribed functions of x.)
As in the case of first order linear equations, in any interval where A(x) ≠ 0, we can replace such an equation
by an equivalent one in standard form:
2. Homogeneous vs. Non-homogeneous Linear Differential
In the development that follows it will be important to distinguish between the case when the right hand
is zero or non-zero. We shall say that a second order
linear ODE is homogeneous if it can be written in
otherwise (if g(x) ≠ 0) we shall say that it is
non-homogeneous. Note that this terminology is completely
unrelated to homogeneous equations of degree zero (the topic of the preceding lecture).
3. Differential Operator Notation
Consider the general second order linear differential equation
We shall often write differential equations like this as
where L is the linear differential operator
That is to say, L is the operator that acts on a function Ø by
4. General Theorems
The following theorem tells us the conditions for the existence and uniqueness of solutions of a second order
linear differential equation.
Theorem 14.1. If the functions p, q and g are continuous on an open interval I R containing the point
, then in some interval about there exists a unique solution to the differential equation
satisfying the prescribed initial conditions
Note how this theorem is analogous to the corresponding
theorem for first order linear ODE's. Note also
that the conditions for existence and uniqueness are fairly lax - all we require is the continuity of the
functions p, q, and g around a given initial point. Finally, we note that the form of the initial conditions
involves the specification of both y(x) and its derivative y'(x) at an initial point
I should also point out that the preceding theorem does not address the issue of how to construct a solution
of a second order linear ODE. Indeed, the actual construction of solutions to second order linear ODE is
sufficiently complicated to that we shall spend 90% of the remaining lectures on techniques of solution. The
next two theorems at least tell us the basic ingredients for a general solution of a second order linear ODE.
Theorem 14.2. (The Superposition Principle) If y = (x) and y = (x) are two solutions of the differential
then any linear combination
of (x) and (x), where
and are constants, is also
a solution of (14.12).
The fact that a linear combination of solutions of a
linear, homogeneous differential equation is
also a solution is extremely important. The theory of linear homogeneous equations, including differential
equations involving higher derivatives depends strongly on the superposition principle
are both solutions of
It is easy to check that any linear combination of and
is also a solution.
are both solutions of
However, it is easy to check that
is not a solution of (14.20). The reason for
this lies in
the fact that (14.20) is not linear.
Given two solutions and of a second order linear homogeneous differential equation
we can construct an infinite number of other solutions
by letting and run through R. The following question
then arises: are all the solutions of (14.21)
capable of being expressed in form (14.22) for some choice of and ?
This will not always be the case; and so we shall say that two solutions and form a fundamental set
of solutions to (14.21) if every solution of (14.21) can be expressed as a linear combination of and
Theorem 14.5. If p and q are continuous on an open interval I = ( α, β) and if and are solutions of
the differential equation
at every point x∈ I, then any other solution of (14.23) on
the interval I can be expressed uniquely as a
linear combination of and .
Let and be two given solutions on an interval I and let Y be an any other solution on I. Choose a
point ∈I. From our basic uniqueness and existence theorem (Theorem 3.2), we know that there is only
solution y(x) of (14.23) such that
namely, Y (x). Therefore if we can show that a solution of the form
satisfies the initial conditions (14.25), then we must
have and so Y (x) is a linear
combination of (x) and (x).
Thus, we now seek to define constants and so that
these initial conditions can be matched. We thus
This is just a series of two equations with two unknowns. Solving the first equation for yields
Plugging this into the second equation yields
Plugging this expression for into (14.27) yields
Thus, we can solve for and whenever the denominator
does not vanish. Thus, so long as and satisfy
(14.23) we can always express any solution as a linear
combination of and .
Remark: The quantity
is called the Wronskian of and .
Example 14.6. Show that
are form a set of fundamental solutions to the differential equation
We simply have to check that the Wronskian does not vanish:
Since the Wronskian does not vanish, and must be linearly independent.