Lecture Notes in Differential Equations - Bruce E. Shapiro

148 LESSON 17. SOME SPECIAL SUBSTITUTIONS
matrix). Then we are allowed to multiply D by y on the right, to gives us
another function. Like the matrix an vector, we are not allowed to reverse
the order of the D and whatever it operates on. Some authors write D[y],
D(y), D t y, or ∂ t y instead of Dy.
Before we begin our study of higher order equations, we will look at what
the D operator represents in terms of linear first order equations. While
it doesn’t really add anything to our understanding of linear first order
equations, looking at how it can be used to describe these equations will
help us to understand its use in higher order linear equations.
We begin by rewriting the first order linear differential equation
as
dy
+ p(t)y = q(t) (17.47)
dt
Dy + p(t)y = q(t) (17.48)
The trick here is to think like matrix multiplication: we are still allowed
the distributive law, so we can factor out the y on the left hand side, but
only on the right. In other words, we can say that
[D + p(t)]y = q(t) (17.49)
Note that we cannot factor out the y on the left, because
so it would be incorrect to say anything like
Dy ≠ yD (17.50)
y(D + p(t)) = q(t) (17.51)
In fact, anytime you see a D that is not multiplying something on its right,
that should ring a bell telling you that something is wrong and you have
made a calculation error some place.
Continuing with equation 17.49 we can now reformulate the general initial
value problem as
}
[D + p(t)]y = q(t)
(17.52)
y(t 0 ) = y 0
The D operator has some useful properites. In fact, thinking in terms of
matrices, it would be nice if we could find an expression for the inverse of
D so that we could solve for y. If M is a matrix then its inverse M −1 has
the property that
MM −1 = M −1 M = I (17.53)