ODE with separate variables

Hi all,

here is one ODE I solved now

[tex]
y' = 1 + y^2
[/tex]

So

[tex]
\frac{y'}{1+y^2} = 1
[/tex]

[tex]
\int \frac{dy}{1+y^2} = \int 1 dx
[/tex]

[tex]
\arctan y = x + C \leftrightarrow y = \tan (x + C)
[/tex]

[tex]
x \in (-\frac{\pi}{2} - C, \frac{\pi}{2} - C)
[/tex]

The last line is what I'm unsure about.

Shouldn't it rather be

[tex]
x \in (-\frac{\pi}{2} - C + k\pi, \frac{\pi}{2} - C + k\pi), k \in \mathbb{Z}
[/tex]

or is it ok as I wrote it originally?

Thank you.
 

StatusX

Homework Helper
2,564
1
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.
 

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