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 proofs.
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 is, 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 .The next theorem is analogous to Theorem thmtype:5.3.1.
Homogeneous Equations
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 nontrivial.
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 dependent onThe next theorem is analogous to Theorem thmtype:5.1.3.
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).
The Wronskian
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 generalizes Theorem thmtype:5.1.4.
Formula (eq:9.1.15) is Abel’s formula.
The next theorem is analogous to Theorem thmtype:5.1.6.
- Proof
- Since are not all zero, (eq:9.1.14) implies that
so
and Theorem thmtype:9.1.4 implies the stated conclusion.
General Solution of a Nonhomogeneous Equation
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 .
The next theorem is analogous to Theorem thmtype:5.3.2.
We’ll apply Theorems thmtype:9.1.6 and thmtype:9.1.7 throughout the rest of this chapter.
Text Source
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)