Expert: Paul Klarreich - 2/25/2006

Question
I have a question about the constant of integration when integrating both sides of a diff e. The equation is:

d
--- ln |p-900| = (1/2)
dt              

The author of the textbook I am reading integrates both sides, obtaining

ln |p - 900| = (t/2) + C

My question is:  Why is one allowed to omit the constant on the left side of the equation?

Thanks.

Answer
Hi, Brian,

You wrote:
Subject:  Differential Equations
Question:  I have a question about the constant of integration when integrating both sides of a diff e. The equation is:

d
-- ln |p-900| = (1/2)
dt          

The author of the textbook I am reading integrates both sides, obtaining

ln |p - 900| = (t/2) + C

My question is: Why is one allowed to omit the constant on the left side of the equation?

Thanks.
-------------------------------------

Because if you wrote:

ln |p - 900| + C1 = (t/2) + C2

It wouldn't be any different.  Writing  f(x) + C  means you are writing not one, but many function all having a general format.  So  f(x) + 17  and  f(x) - 3  are some of these examples.

And if you wrote, for example

ln |p - 900| + 17 = (t/2) + 29

instead of

ln |p - 900|  = (t/2) + 12

it is clearly the same example.  So there is no need to put a constant term symbol on both sides -- one of them will do.