# Diff equation: arctan

• Dec 6th 2007, 07:11 PM
paolopiace
Diff equation: arctan
I'm not clear yet how to write the math symbols.
Therefore I'm wirting in plain text.

The separable differential equation I'm dealing with ends up as:

arctan(y) = arctan(x) + K

How do I find the solutions from here?

Thanks!
• Dec 6th 2007, 07:49 PM
topsquark
Quote:

Originally Posted by paolopiace
I'm not clear yet how to write the math symbols.
Therefore I'm wirting in plain text.

The separable differential equation I'm dealing with ends up as:

arctan(y) = arctan(x) + K

How do I find the solutions from here?

Thanks!

It would probably be better to leave the equation "as is" and figure out a value of K from initial conditions. But then you simply have
$y = tan(tan^{-1}(x) + K)$

The possible problem I see here is that $tan^{-1}(x)$ is going to return a reference angle, so technically your solution will be
$y = tan(tan^{-1}(x) + \pi n + K)$
where n is an integer.

I suppose the source (as it were) of the differential equation will have to tell you how to handle that.

I'm not sure if this answers your question....

-Dan
• Dec 6th 2007, 09:08 PM
paolopiace
Thanks topsquark - you gave me the hint...
Here is how I did:

y = tan(arctan(x) + K) = [x + tan(K)]/[1-x*tan(K)] = (x + C)/(1-x*C)

That's exactly what I was looking for!
• Dec 7th 2007, 07:33 AM
topsquark
Quote:

Originally Posted by paolopiace
Here is how I did:

y = tan(arctan(x) + K) = [x + tan(K)]/[1-x*tan(K)] = (x + C)/(1-x*C)

That's exactly what I was looking for!

Ah! I had never thought to use the sum of angles formula. (Handshake)

-Dan