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

[GreenAce92]

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 ?

 

[UltrafastPED]

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.

 

[HallsofIvy]

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 "$\phi$".

 

[GreenAce92]

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.

 

[blue_raver22]

I can say that the answer to the first question is yes, you can split up the left hand side of an ODE. In fact, this is a common technique used in solving ODEs, known as separation of variables. However, it is important to note that this can only be done in certain cases and it is not a guaranteed method for solving all ODEs.

Regarding the second question, it is not safe to assume that all problems involving integrating factors will result in a perfect integral pair. While this may be the case in some simpler problems, it is not always the case and it is important to carefully consider the specific ODE and its characteristics before making any assumptions.

In cases where the left hand side is not easy to differentiate, there are other techniques that can be used to solve the ODE. These may include substitution, transformation, or using more advanced methods such as power series or Laplace transforms. It is important to have a strong understanding of the various techniques and their limitations in order to effectively solve ODEs.