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When solving elementary, separable ordinary differential equations (ODEs), and obtaining the original family of solutions:

  1. How do I know the domain over which the family of solutions satisfies the differential equation?

  2. Does it satisfy it for the entire $\Bbb R$?

  3. If so, can I always find a particular solution going through any point on the $xy$ plane?

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The domain of the family solutions depends on the specific function that solves your ODE.

For example the solution of $$xy'=1$$ is $$y=\ln|x|+C$$ where $C$ is any real number. Here you won't have any solution points going through the left half of the plane and on the $x=0$ line. Thus $D_y:0<x<\infty$

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  • $\begingroup$ But naturally when I find an analytic solution using one of the techniques, when there isn't any domain issues, then it's valid for all values of x right? $\endgroup$ Apr 19, 2022 at 11:02
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    $\begingroup$ I am not sure what you mean by analytic in this context, the domain of the solution (function) always depends on the solution (function). If there are no 'domain issues', then it's valid for all real numbers, $\forall x\in \mathbb{R}$. It might be an issue in terms of understanding the solution space, but this is just the nature of the functions. Kind of like saying $\frac{1}{x}$ takes up no values at $x=0$. It is given. $\endgroup$ Apr 19, 2022 at 19:44
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    $\begingroup$ Yeah got it! thanks! $\endgroup$ Apr 19, 2022 at 19:46

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