# The General, Linear Equation

**2.3 The first-order system
**

We introduce in this section a somewhat different formulation of linear differential

equations. In fact, the formulation applies to nonlinear differential

equations as well, but we restrict our present considerations to the linear

case. We begin by considering the n-th order linear equation (2.4).

**2.3.1 The system equivalent to the n-th order equation**

There is an alternative representation of the n-th order linear equations (2.3)

and (2.4); we’ll concentrate on the latter since it contains the former if r ≡ 0.

Define

The equation (2.4) is now equivalent to the system of n first-order equations

If initial-data are provided in the form indicated in
equation (2.31), then

for i = 1, . . . , n.

This has a compact expression in vector-matrix notation. Denote by v

the column vector

and by A the matrix

The inhomogeneous equation then has the expression

where v' is the vector made up of the corresponding
derivatives of the components

of v, and

**2.3.2 The general, linear system
**

The form of equation (2.40) looks, perhaps superficially, like that of the general

linear equation of first order encountered in the first chapter. It turns out

that the resemblance is more than superficial. Procedures from the abstract

(like the existence theorem) to the practical (like numerical solutions) are

unified by this formulation. Moreover, there is no need to restrict the first-order

systems of n equations that we consider to those derived from a single,

n-th order equation: we are free to consider the more general case of equation

(2.40) above wherein the matrix is an arbitrary matrix of

continuous functions, rather than one having the special companion-matrix

structure of equation (2.39) above, and the vector

is an arbitrary vector of continuous functions rather than having the special

structure of equation (2.38).

Consider again the initial value-problem

where A(x) is an n×n matrix whose entries are continuous
functions on an

interval I, R(x) is an n-component vector whose components are continuous

functions on I, is a point of I, and
is an arbitrary vector of constants.

We again appeal to Chapter 6 for the basic conclusion:

**
Theorem 2.3.1 **The initial-value problem (2.42) possesses a unique solution

on I.

This means that there is a vector v whose components are differentiable

functions of x on I, satisfies the differential equation v' = Av + R at each

point of I, and reduces to the given vector when x = . Again, as in the

case of the single equation of order n, it is useful to consider the homogeneous

case R ≡ 0 first.

**The homogeneous system
**

The homogeneous system is written in our current vector notation as

If v and w are both vector solutions of this equation, so
also is

for arbitrary constants α and β. This leads as before to the notion of linear

dependence of vector-valued functions on an interval I:

**
Definition 2.3.1** The k n-component vector functions
are

said to be linearly dependent on the interval I if there exist k real numbers

, not all zero, such that

each point x of I. Otherwise they are said to be linearly
independent.

In the theory of finite-dimensional vector spaces (like R^{n}, for example),

a set of k n-component vectors is likewise said to be

linearly dependent if a relation holds for scalars

that are not all zero. The vectors
above are functions

on an interval I and belong, for any particular value in I

to the n-dimensional vector space R^{n}. But the requirement of linear

dependence is more stringent under the definition 2.3.1 than the

corresponding definition for a finite-dimensional vector space. The

relation

might hold for many values
with a set of constants that
are not all

zero (i.e., they may be linearly dependent as vectors in R^{n} at those

points) without implying linear dependence on I: for this the equation

above would have to hold, for a fixed set of constants
at

every point of I.

For example, consider the vector functions and

on the interval [−1, 1] (the superscript t stands for

transpose). It is easy to see that these are linearly independent under

the definition 2.3.1. However, if we set x = 0, each of them is the zero

vector and they are linearly dependent as vectors in R^{2}.

It is clear from the definition that, for any set of k
functions

defined on the interval I, linear dependence on I implies linear dependence

on R^{n} for each x ∈ I. Since linear independence is the negation of linear

dependence, we also have linear independence on R^{n} at any point of I implies

linear independence on I.

We can easily infer the existence of a set of n linearly independent solutions

of the homogeneous equation: it suffices to choose them to be linearly

independent as vectors in R^{n} at some point ∈ I. For example, we could

choose ()
= (1, 0, . . . , 0)^{t} to be the first unit vector,

to be the second unit vector, and so on. We now show that the solutions

of the homogeneous equation with these initial data

are linearly independent according to the definition 2.3.1. For suppose that,

for some value , the n vectors
are linearly
dependent

as vectors in R^{n}. Then there are constants
, not all zero,

such that

Using these constants, denote by v(x) the sum

It is a solution of the homogeneous equation vanishing at
. But by the

uniqueness theorem it must vanish identically on I. This is not possible at

unless
, a contradiction. Since
could have been any

point of I, this shows that
is linearly independent on R^{n} at any point

of I and therefore, by the remark above, linearly independent on I. We have

proved the following theorem:

**Theorem 2.3.2** Let each of the n-component vector functions
satisfy

the homogeneous differential equation (2.43), and suppose that for some

∈ I the vectors
are linearly independent as vectors in R^{n}. Then

they are linearly independent on I.

As before, a set of linearly independent solutions of the homogeneous

equation is called a basis of solutions, and any solution of the homogeneous

equation can be written as a linear combination of them. If
is a

basis of solutions, one can form the matrix

This matrix, sometimes called a fundamental matrix
solution of equation

(2.43), and consisting of columns of a basis of vector-valued solutions,
satisfies

the matrix version of the latter equation,

and is nonsingular at each point x of I. A useful version
of this matrix is

that which reduces to the identity matrix E at a specified point . In terms

of this matrix, the solution v(x) of the homogeneous problem reducing to the

vector when x = is
.

**The Inhomogeneous Solution
**

Return now to the inhomogeneous problem (2.42), and suppose that we have

solved the homogeneous problem and can form a fundamental matrix solution

. The variation-of-parameters formula has the structure ;

we leave it to the reader (problem 12 below) to work out the function w such

that this provides a solution of the inhomogeneous problem.

PROBLEM SET 2.3.1

PROBLEM SET 2.3.1

1. Work out the Wronskian for the solutions found in Example 2.2.2. Referring

to Theorem 2.2.2, explain why the Wronskian is constant.

2. Same problem as the preceding for the solutions of Example 2.2.3.

3. Find a basis of solutions for the system u'''+u'' = 0. Calculate theWronskian

and check the result against Theorem 2.2.2 (equation 2.33).

4. Same as Problem 3 but for the equation

5. Write out the Wronskian W
for three
functions. Assuming that

they satisfy a third-order, linear, homogeneous differential equation, derive

the formula of Theorem 2.2.2.

6. The formula for the derivative of a determinant whose
entries depend on a

variable x is

that is, it consists of n determinants, of which the first
is obtained by differentiating

the entries in the top row and leaving the others unchanged, the

second by differentiating the entries of the second row and leaving the others

unchanged, etc. Using this and other, more familiar, rules for manipulating

determinants, prove the general version of Theorem 2.2.2. tex

7. For the operator Lu = u'''+u', find the most general solution of the equation

Lu = 1 (cf. Example 2.2.3).

8. For the operator of the preceding problem, obtain the influence function for

solving the inhomogeneous problem.

9. Carry out the uniqueness part of Theorem 2.2.1 if n=3.

10. Carry out the proof of Theorem 2.1.4 for the nth-order homogeneous problem.

11. Find the equivalent first-order system (that is, find the matrix A and the

vector R of equation (2.40)) for the second-order equation

12. In the general, inhomogenous system v' = A(x)v +
R(x), choose to be a

fundamental matrix solution of the homogeneous problem that reduces to

the identity at x = . Make the substitution v(x) =
(x)w(x) to find a

particular integral v that vanishes at the point of I.

13. Let represent the determinant of an n×n matrix solution of equation

(2.46) above. Using the formula given in problem 6 above, show that

satisfies a first order equation

and express the function a(x) in terms of the entries of the matrix A.