Can I split up the left hand side of an ODE?

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Discussion Overview

The discussion revolves around the use of integrating factors in solving ordinary differential equations (ODEs), particularly whether one can assume that the left-hand side of an ODE will always yield a straightforward integral when an integrating factor is applied. Participants explore the implications of this assumption and the challenges faced in applying these techniques in both academic and real-world contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant questions whether problems involving integrating factors will always lead to a "perfect integral," suggesting uncertainty about the terminology and the conditions under which this holds true.
  • Another participant emphasizes that various techniques apply to specific classes of linear ODEs and cautions against expecting a single technique to work universally.
  • A different participant clarifies that while both sides of an equation are typically continuous and integrable, the ability to perform the integral depends on the individual's skill and understanding.
  • Concerns are raised about the nature of learning in differential equations, with one participant expressing frustration over a perceived lack of intuitive understanding and reliance on rote application of methods.
  • There is acknowledgment that while textbook problems may be simplified, real-life applications often require numerical solutions, complicating the learning process.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions surrounding integrating factors and their outcomes. There is no consensus on whether one can universally expect a "perfect integral" from the left-hand side of an ODE when using integrating factors, and the discussion reflects a range of experiences and interpretations regarding the learning process in differential equations.

Contextual Notes

Participants highlight limitations in their understanding and the challenges of applying theoretical methods to practical problems. There is a recognition of the potential disconnect between textbook learning and real-world applications.

GreenAce92
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Should I just assume that any problems that involve integrating factor will always result in a perfect integral pair? That's probably not the right terminology but for instance if I have a differential equation which has had an integrating factor multiplied to both sides, then the left hand side most likely becomes a 'perfect integral' as an arbitrary example, a left hand side is y' e^2t + 2y e^2t which if I integrate this as a whole, I would say well that is y e^2t.

Can I assume that this will always be the case in terms of an entry level differential equations class? What happens if the left hand side is not easy to differentiate ?
 
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You will learn a number of techniques; each will apply to a certain class of linear ordinary differential equations.

You should not expect a particular technique to work on every ODE.
 
GreenAce92 said:
Should I just assume that any problems that involve integrating factor will always result in a perfect integral pair?
What do you mean by "perfect integral"? In all except "pathological cases", both sides will be continuous which will certainly be "integrable". But I suspect that you mean "An integral that I can do" and that, of course, depends upon you!

(I remember in my very first d.e. course, first chapter, I was able to reduce a d.e. in a homework problem to an integral but just wasn't able to do the integral. The solution to the very next problem in the text was in the back of the book so I looked- and found that the answer was given in terms of an integral!)

That's probably not the right terminology but for instance if I have a differential equation which has had an integrating factor multiplied to both sides, then the left hand side most likely becomes a 'perfect integral' as an arbitrary example, a left hand side is y' e^2t + 2y e^2t which if I integrate this as a whole, I would say well that is y e^2t.
If the problem is a "book home work" problem the either it is one that has been "made up" to be comparatively easy or you are expected to leave it as an integral. If you are talking about "real life", engineering problems, then "almost all" differential equations can be solved only numerically.

Can I assume that this will always be the case in terms of an entry level differential equations class? What happens if the left hand side is not easy to differentiate ?

Did you mean to say "differentiate" here? You want to integrate to find the solution. And the whole point of an "integrating factor" is to get the left hand side in the form "d\phi" which has the obvious integral "\phi".
 
Thank you for the responses, reading my own posts it would seem clear that I have not been doing my work thus I don't even know what words I am using.

I have been studying and the material has been making sense

Still I find that lately I haven't had a good learning experience, specifically to me, I think that learning has become "this is what you want to do, find this on the test..." it's not really on "intuition" or instinct

When I read a real life problem like a lake draining and filling, or salt dissolving, it seems to make sense but at the same time, these methods seem like I am just expected to apply them. I can't say that the proof wasn't shown to us, I think the books cover the proofs of why these methods are valid...

Anyway, thank you for your time.
 

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