What is the Correct Domain for this ODE Solution?

[twoflower]

Hi all,

here is one ODE I solved now

<br /> y&#039; = 1 + y^2<br />

So

<br /> \frac{y&#039;}{1+y^2} = 1<br />

<br /> \int \frac{dy}{1+y^2} = \int 1 dx<br />

<br /> \arctan y = x + C \leftrightarrow y = \tan (x + C)<br />

<br /> x \in (-\frac{\pi}{2} - C, \frac{\pi}{2} - C)<br />

The last line is what I'm unsure about.

Shouldn't it rather be

<br /> x \in (-\frac{\pi}{2} - C + k\pi, \frac{\pi}{2} - C + k\pi), k \in \mathbb{Z}<br />

or is it ok as I wrote it originally?

Thank you.

 

[StatusX]

It depends. Usually you would determine the constant by an initial value, ie, y(x₀)=y₀, in which case the solution would be valid within whichever region (ie, of width \pi[/tex]) contains x₀. If you&#039;re just looking for a completely general form, one that represents every possible solution, then your second form is correct.