We study the theory of homogeneous linear systems, noting the parallels with the study of linear homogeneous scalar equations.
Basic Theory of Homogeneous Linear System
In this section we consider homogeneous linear systems , where is a continuous matrix function on an interval . The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations, which we considered in Trench 2.1, 5.1, and 9.1.
Whenever we refer to solutions of we’ll mean solutions on . Since is obviously a solution of , we call it the trivial solution. Any other solution is nontrivial.
If are vector functions defined on an interval and are constants, then
is a linear combination of . It’s easy show that if are solutions of on , then so is any linear combination of We say that is a fundamental set of solutions of on on if every solution of on can be written as a linear combination of , as in (eq:10.3.1). In this case we say that (eq:10.3.1) is the general solution of on .It can be shown that if is continuous on then has infinitely many fundamental sets of solutions on The next definition will help to characterize fundamental sets of solutions of .
We say that a set of -vector functions is linearly independent on if the only constants such that
are . If (eq:10.3.2) holds for some set of constants that are not all zero, then is linearly dependent onThe next theorem is analogous to Theorems thmtype:5.1.3 and thmtype:9.1.2.
We can use the method in Example example:10.3.1 to test solutions of any system for linear independence on an interval on which is continuous. To explain this (and for other purposes later), it’s useful to write a linear combination of in a different way. We first write the vector functions in terms of their components as If then
and
that is, the columns of are the vector functions .For reference below, note that
The determinant of ,
is called the Wronskian of . It can be shown that this definition is analogous to definitions of the Wronskian of scalar functions given in Trench 5.1 and 9.1. The next theorem is analogous to Theorems thmtype:5.1.4 and thmtype:9.1.3.The next theorem is analogous to Theorems thmtype:5.1.6 and thmtype:9.1.4.
We say that in (eq:10.3.4) is a fundamental matrix for if any (and therefore all) of the statements item:10.3.3a-item:10.3.3e of Theorem thmtype:10.3.2 are true for the columns of . In this case, (eq:10.3.3) implies that the general solution of can be written as , where is an arbitrary constant -vector.
- (a)
- Compute the Wronskian of directly from the definition (eq:10.3.5)
- (b)
- Verify Abel’s formula (eq:10.3.6) for the Wronskian of .
- (c)
- Find the general solution of (eq:10.3.7).
- (d)
- Solve the initial value problem
item:10.3.2b Here so tr. If is an arbitrary real number then (eq:10.3.6) implies that which is consistent with (eq:10.3.9).
item:10.3.2c Since , Theorem thmtype:10.3.3 implies that is a fundamental set of solutions of (eq:10.3.7) and is a fundamental matrix for (eq:10.3.7). Therefore the general solution of (eq:10.3.7) is
item:10.3.2d Setting in (eq:10.3.10) and imposing the initial condition in (eq:10.3.8) yields Thus,
Text Source
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)