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I am a first year Master's student in applied mathematics. I'm currently taking a Mathematical Physics class and we have been studying techniques for solving ODEs. There is some overlap with undergrad ODEs.

Solving these problems (ex: $x^2y''+xy'-16y=8x^4$) often takes lots of time and lots of algebraic manipulation, all to find a solution that I won't use.

I understand that it is important to become familiar with these equations and have an idea about how they are solved. But I am failing to see the big picture.

How will knowing myriad solution techniques for these ODEs help my future research in applied mathematics?

Have you found this knowledge useful in your research?

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    $\begingroup$ This famous essay by Gian Carlo Rota can be an interesting read for you. $\endgroup$ – Giuseppe Negro Dec 3 '17 at 18:26
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    $\begingroup$ "Myriad solution techniques" : maybe 50 or 100 years ago... But nowadays, it"s far from that, in particular with computers. You well never be an "applied mathematician" without a good background in this very fundamental (and rewarding) field... $\endgroup$ – Jean Marie Dec 3 '17 at 18:27
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    $\begingroup$ @GiuseppeNegro That was a very fascinating read. Unfortunately, it damps my motivation to study for the final exam! :) $\endgroup$ – EternusVia Dec 3 '17 at 18:58
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    $\begingroup$ @EternusVia: Dang! :-) I am sorry. The opposite is true: a solid knowledge of ODEs is something that I would have found very useful during my PhD studies. $\endgroup$ – Giuseppe Negro Dec 3 '17 at 19:02
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Here's an example of a problem I encountered during my PhD studies in which my lack of knowledge of ODEs turned out to be a burden. As you see in comments, the right way to approach such things requires the use of special functions. (Which are something that all applied mathematicians, and many pure mathematicians, need to deal with quite often). But how do special functions arise? As you see in the very first line of the linked Wikipedia article (section "Overview"), they are always the solution to some ODE. (The page says "often", I changed the adverb according to my experience).

You might still wonder why you should spend time transforming ODEs, playing with ill-defined differentials, doing endless frustrating computations like the ones you mention. I wished I had in many occasions, since that train of ideas is the main one that led Sophus Lie to formulate his theory. We're talking about symmetry, which in my experience is the only idea that is common to all mathematics, physics, and engineering. If you are not fluent with that kind of computations, like me, you are like a lion without teeth.

I can keep talking about this for an hour, if you want.

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  • $\begingroup$ A good book to read about symmetry and PDEs is "symmetries of differential equations" by bluman. With this book, all of the 'tricks' of differential equations can start to make sense in a unified way. Indeed, symmetry is the 'correct' way to study differential equations (unless you're more into numerical computing). $\endgroup$ – Tony S.F. Jan 15 '18 at 16:08
  • $\begingroup$ A good book to read about symmetry and PDEs is "symmetries of differential equations" by bluman. With this book, all of the 'tricks' of differential equations can start to make sense in a unified way. Indeed, symmetry is the 'correct' way to study differential equations (unless you're more into numerical computing). $\endgroup$ – Tony S.F. Jan 15 '18 at 16:08

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