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With x and y being independent variables, if:

$\frac{\partial f(x,t)}{\partial t} = \frac{\partial g(x,t)}{\partial x}$

Is it correct to assume that:

$\int\frac{\partial f(x,t)}{\partial t}dt = \int\frac{\partial g(x,t)}{\partial x}dx \iff f(x,t) = g(x,t)$

?

BCLC
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c.leblanc
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    It's rather obviously not correct. Have you tested your theory on some simple examples? – Hans Lundmark Nov 13 '22 at 21:56
  • You are right, I have seen this kind of things with finite integrals in EDP solving though. I am trying to understand how and when exactly we can do this kind of thing. – c.leblanc Nov 13 '22 at 22:42
  • Maybe you're thinking of the method of separation of variables for ordinary differential equations? That's just the chain rule backwards; see for example this question: https://math.stackexchange.com/questions/27425/what-am-i-doing-when-i-separate-the-variables-of-a-differential-equation. – Hans Lundmark Nov 14 '22 at 05:25
  • It was in wave equations solving using the caracteristic method for 2nd order EDP, but I think I made a huge shortcut in the way I asked my question. I'll upload a more precise example when I'm back home later today. – c.leblanc Nov 14 '22 at 08:30
  • OK. (By the way, in English the abbreviation is PDE rather than EDP.) – Hans Lundmark Nov 14 '22 at 09:55

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