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I've been doing excercises for first order differential equations, there are like 7 methods so far and I wonder, what if one day i find a differential equation which doens't fit in any method? I'm not sure I can prove those exist, if they do, how would I solve it?

José Osorio
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  • What methods are you speaking of? Methods of solving the equations? And do you mean general 1 order equations or maybe of the form $y'=f(x,y)$? – Yuriy S Jun 12 '16 at 08:57
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    Most differential equations cannot be solved by standard methods. Only way to do it is using some computer approximation. The only ones you are able to solve are the ones you find as exercises in the textbook :-) – Ant Jun 12 '16 at 08:58
  • Unless the First Order DE is very, very awkward, the method described here should work. – Git Gud Jun 12 '16 at 09:30

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Depends on the context :

  • If it's an exercise then either there's a method you don't know or there's a way to solve it specific to this particular equation/exercise which you should be able to find on your own

  • If it's a "real life" problem (e.g. engineering problem) the you can try and prove some properties of the solution (is it periodic, is it even,...) and you'll most likely end up solving it numerically.

pmichel31415
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There are certainly differential equations which don't fit any of the $7$ methods given in any textbook. There are still many famous unsolved problems in differential equations. From the Clay's Mathematics Institute "Millennium Problems" list:

Navier–Stokes Equation

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

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Ovi
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