What is the Correct Domain for this ODE Solution?

[twoflower]

Hi all,

here is one ODE I solved now

$$y' = 1 + y^2$$

So

$$\frac{y'}{1+y^2} = 1$$

$$\int \frac{dy}{1+y^2} = \int 1 dx$$

$$\arctan y = x + C \leftrightarrow y = \tan (x + C)$$

$$x \in (-\frac{\pi}{2} - C, \frac{\pi}{2} - C)$$

The last line is what I'm unsure about.

Shouldn't it rather be

$$x \in (-\frac{\pi}{2} - C + k\pi, \frac{\pi}{2} - C + k\pi), k \in \mathbb{Z}$$

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 [itex]\pi[/tex]) contains x₀. If you're just looking for a completely general form, one that represents every possible solution, then your second form is correct.