Lecture Notes in Differential Equations - Bruce E. Shapiro

4 LESSON 1. BASIC CONCEPTS
From our study of calculus we know that
∫ dy
y
= ln |y| + C (1.16)
and
∫
dt = t + C (1.17)
where the C’s in the last two equations are possibly different numbers. We
can write this as
ln |y| + C 1 = t + C 2 (1.18)
or
where C 3 = C 2 − C 1 .
ln |y| = t + C 3 (1.19)
In general when we have arbitrary constants added, subtracted, multiplied
or divided by one another we will get another constant and we will not
distinguish between these; instead we will just write
ln |y| = t + C (1.20)
It is usually nice to be able to solve for y (although most of the time we
won’t be able to do this). In this case we know from the properties of
logarithms that a
|y| = e t+C = e C e t (1.21)
Since an exponential of a constant is a constant, we normally just replace e C
with C, always keeping in mind that that C values are probably different:
|y| = Ce t (1.22)
We still have not solved for y; to do this we need to recall the definition of
absolute value:
{ y if y ≥ 0
|y| =
(1.23)
−y if y < 0
Thus we can write
y =
{
Ce
t
if y ≥ 0
−Ce t if y < 0
(1.24)
But both C and −C are constants, and so we can write this more generally
as
y = Ce t (1.25)
So what is the difference between equations 1.22 and 1.25? In the first case
we have an absolute value, which is never negative, so the C in equation