These systems of equations are just like what was discussed earlier which included x^{′} = ax + f

(t)

. The only
difference is that here we consider

x′ = Ax + f (t)

where A is an n × n matrix. It turns out these equations are completely routine and that once you have
these understood, you also understand higher order scalar linear differential equations of the
form

y(n) + an−1y(n−1) + ⋅⋅⋅+ a1y′ + a0y = f (t)

Indeed, such an equation can be written as a first order system as follows.

( x1 )′ ( x2 )
| x | | x |
|| .2 || || 3. ||
|| .. || = || .. ||
|( xn−1 |) |( xn |)
x − (a x + ⋅⋅⋅+ ax + a x)
n n− 1 n 12 0 1

When we take the derivative of a matrix whose entries are functions of t the symbol simply means to
take the derivative of all the entries. Thus

( ) ( )
et sint ′ et cost
t2 cost ≡ 2t − sint

Before doing anything else, here is a fundamental observation about the product rule for products
of matrices whose entries are functions of t. Thus A_{ij}

(t)

is the ij^{th} entry of A

(t)

an m × n
matrix.

Theorem 24.0.1Suppose A

(t)

and B

(t)

are matrices whose entries are differentiable functions of t.Suppose also that A

(t)

B

(t)

makes sense. Then

(A (t)B(t))′ = A ′(t)B (t) + A(t)B′(t)

Proof: This follows directly from the above definition of what it means to differentiate a matrix which
is a function of t.

( ) ( )′
(A(t)B (t))′ij ≡ (A (t)B (t))ij

( ) ′
≡ ∑ A (t) B(t) produ=ct rule ∑ A′(t) B (t) + A (t) B′(t)
k ik kj k ik kj ik kj
≡ (A′(t)B (t)+ A (t)B ′(t))
ij

which shows that the product rule holds as claimed. ■

Note how you must be careful to keep the order the same since matrix multiplication is not
commutative. Other than that, it is just the ordinary product rule.

Example 24.0.2Write y^{′′′} + y^{′′}− 2y^{′} + 3y = sint as a first order system.

This is a third order equation having highest derivative the third derivative and so it will be
represented by a first order system involving a 3 × 3 matrix. Then as a first order system, it
is

and the solution to the original higher order system is given by x_{1}.

When you have a system

′
x = Ax + f

the initial condition should be of the form x

(a)

= x_{0}. In the case of a first order system which comes from
a higher order scalar equation of order n, this means you should give the initial condition on the function,
and its first n − 1 derivatives.

Although you can always write a higher order linear equation as a first order system of equations, it
does not go the other way around. Therefore, it really is pointless to waste a lot of time on
higher order scalar linear differential equations. The exception to this is the case of linear scalar
differential equations in which you have variable coefficients and which cannot be placed in the form
described above. Then it is appropriate and necessary to consider special methods specific to these
equations.