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Given an th order linear differential equation, we discuss necessary and sufficient
conditions for a set of functions to be a fundamental set of solutions.
Introduction to Linear Higher Order Equations
An th order differential equation is said to be linear if it can be written in the form
We considered equations of this form with in Trench 2.1, and with in Trench 5.1 and
5.3. In this chapter is an arbitrary positive integer.
In this section we sketch the general theory of linear th order equations. Since this
theory has already been discussed for in Trench 5.1 and 5.3, we’ll omit
For convenience, we consider linear differential equations written as
which can be rewritten as (eq:9.1.1) on any interval on which has no zeros, with , …, and .
For simplicity, throughout this chapter we’ll abbreviate the left side of (eq:9.1.2) by ; that
We say that the equation is normal on if , , …, and are continuous on and has no
zeros on . If this is so then can be written as (eq:9.1.1) with and continuous on
Suppose is normal on , let be a point in and let , , …, be arbitrary real numbers.
Then the initial value problem
has a unique solution on .
Eqn. (eq:9.1.2) is said to be homogeneous if and nonhomogeneous otherwise. Since is
obviously a solution of , we call it the trivial solution. Any other solution is
If are defined on and are constants, then
is a linear combination of . It’s easy to show that if are solutions of on , then so is
any linear combination of . (See the proof of Theorem thmtype:5.1.2.) We say that is a
fundamental set of solutions of on if every solution of on can be written as a linear
combination of , as in (eq:9.1.3). In this case we say that (eq:9.1.3) is the general solution of on
It can be shown that if the equation is normal on then it has infinitely many
fundamental sets of solutions on . The next definition will help to identify
fundamental sets of solutions of .
We say that is linearly independent on if the only constants such that
are . If (eq:9.1.4) holds for some set of constants that are not all zero, then is linearly
If is normal on , then a set of solutions of on is a fundamental set if and only if
it’s linearly independent on .
is normal and has the solutions , , and on and . Show that is linearly independent
on and . Then find the general solution of (eq:9.1.5) on and .
on . We must show that . Differentiating (eq:9.1.6) twice yields the system
If (eq:9.1.7) holds for all in , then it certainly holds at ; therefore,
By solving this system directly, you can verify that it has only the trivial solution
; however, for our purposes it’s more useful to recall from linear algebra
that a homogeneous linear system of equations in unknowns has only the
trivial solution if its determinant is nonzero. Since the determinant of (eq:9.1.8)
it follows that (eq:9.1.8) has only the trivial solution, so is linearly independent on . Now
Theorem thmtype:9.1.2 implies that
is the general solution of (eq:9.1.5) on . To see that this is also true on , assume that (eq:9.1.6)
holds on . Setting in (eq:9.1.7) yields
Since the determinant of this system is
it follows that ; that is, is linearly independent on .
is normal and has the solutions , , , and on . (Verify.) Show that is linearly
independent on . Then find the general solution of (eq:9.1.9).
Suppose , , , and are constants such that
for all . We must show that . Differentiating (eq:9.1.10) three times yields the system
If (eq:9.1.11) holds for all , then it certainly holds for . Therefore
The determinant of this system is
so the system has only the trivial solution . Now Theorem thmtype:9.1.2 implies that
is the general solution of (eq:9.1.9).
We can use the method used in Examples example:9.1.1 and example:9.1.2 to test solutions of any th order
equation for linear independence on an interval on which the equation is normal.
Thus, if are constants such that
then differentiating times leads to the system of equations
for . For a fixed , the determinant of this system is
We call this determinant the Wronskian of . If for some in then the system (eq:9.1.13) has
only the trivial solution , and Theorem thmtype:9.1.2 implies that
is the general solution of on .
The next theorem is analogous to Theorem thmtype:5.3.2. It shows how to find the general
solution of if we know a particular solution of and a fundamental set of solutions of
the complementary equation .
Suppose is normal on . Let be a particular solution of on , and let be a
fundamental set of solutions of the complementary equation on . Then is a solution
of on if and only if
where are constants.