My Math Forum First Order Differential Equations By Separation of Variables & A Few Other Problems

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 April 7th, 2017, 07:36 PM #1 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 First Order Differential Equations By Separation of Variables & A Few Other Problems Hi There, I am new to this forum. I was going through some of my old university notes the other day pertaining to calculus and solving first order differential equations via separation of variables, and came across a problem where some of the simplifications I wrote down at the time (20 years ago) from either myself, or the lecturer, just didn't make sense. The problem probably lies with me; as these days, at work, if I do math, I tend not to do it manually and via software. So, whilst I remember some of the fundamentals and pretty much know how the basics work; I have forgotten some of the simplifications and shortcuts. Anyway, the above problem kind of led me to a few other issues too, and I was hoping that by joining this forum someone could assist and/or explain to me where I have gone wrong and what the real explanations for my confusion are. Would you/anyone mind? If not I will then upload a document I recently wrote on the matter that hopefully explains the problem(s). Kind regards, Jim. Last edited by skipjack; April 7th, 2017 at 11:24 PM.
 April 7th, 2017, 11:29 PM #2 Global Moderator   Joined: Dec 2006 Posts: 17,221 Thanks: 1294 Welcome to the forum. We'll try to help you if you post your questions.
 April 10th, 2017, 01:48 PM #3 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 OK, thanks SJ. I need to make sure I have re-written it out properly first. As whilst the math itself is not overly complex some of the simplifications used combined with my non/directly related issues with it probably overcomplicate it; but since I want to discuss them too I better make sure I have laid it all out properly. I’ll return with the problem and some questions. Cheers, Jim.
 April 10th, 2017, 03:37 PM #4 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 Actually, before I do that. If I may, I will ask this question first. Please bear with me if it seems meaningless, as it will (hopefully) become clearer why I do this once I complete the aforementioned task within my post #3. Question: what function of $\displaystyle \ln$ (natural logarithm) - when derived - equals $\displaystyle 1/( x^2+ 1)$? I am not looking for $\displaystyle \arctan$ $\displaystyle (x)$ As I am aware that $\displaystyle d/dx \arctan(x) = 1/( x^2+ 1)$ and that $\displaystyle ∫ 1/( x^2 + 1)\,dx$ = $\displaystyle \arctan (x)$ I am looking for an antiderivative of $\displaystyle 1/(x^2+ 1)$ that is itself: A) Not a function of $\displaystyle \arctan$ B) An explicit natural logarithm function and expression; preferably without using substitution methods such as and/or similar to; $\displaystyle u= x^2+ 1$ As I know that $\displaystyle ∫ 1/( u)$ $\displaystyle du$ = $\displaystyle \ln$ $\displaystyle u$ + $\displaystyle C$ and that $\displaystyle d/du$ $\displaystyle \ln$ $\displaystyle u$ = $\displaystyle 1/u$ Cheers, Jim. Last edited by skipjack; April 10th, 2017 at 07:47 PM.
 April 10th, 2017, 07:25 PM #5 Global Moderator   Joined: Dec 2006 Posts: 17,221 Thanks: 1294 The answer must be related to arctan($x$) in some way. $\dfrac{1}{1 + x^2} = \dfrac{i}{2}\left(\dfrac{1}{x + i} - \dfrac{1}{x - i}\right)$, where $i = \sqrt{-1}$. Integrating gives $\dfrac{i}{2}(\ln(x + i) - \ln(x - i))$ + C, where C is a constant. Thanks from Benit13
 April 11th, 2017, 03:21 PM #6 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 Thanks for that SJ. I was reasonably sure I was not going to be able to get away from arctan (and/or complex numbers); but just wanted to check with someone that knew more about it than me. As soon as I get time to write the main problem out I’ll do it and return with some more questions. Cheers, Jim.
 April 13th, 2017, 11:12 PM #7 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 OK, now I have another problem. I was just about to upload the equation of question and get you to check the accuracy of it before we continued but . . . The resulting .pdf is 639KB and the site’s upload facility won’t allow me to upload a .pdf file of that size. Additionally, I can't upload the .doc file either, and the calculations I have covered are probably too lengthy to try and neatly/manually type them in the post GUI. Any suggestions? Cheers, Jim.
April 13th, 2017, 11:26 PM   #8
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 Originally Posted by JumboJimBo OK, now I have another problem. I was just about to upload the equation of question and get you to check the accuracy of it before we continued but . . . The resulting .pdf is 639KB and the site’s upload facility won’t allow me to upload a .pdf file of that size. Additionally, I can't upload the .doc file either, and the calculations I have covered are probably too lengthy to try and neatly/manually type them in the post GUI. Any suggestions? Cheers, Jim.
You might have to compress your files.

Also, you can start a new thread when you do.

 April 13th, 2017, 11:46 PM #9 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 Thanks for that, but it still exceeds the limits when compressed. Cheers, Jim.
 April 18th, 2017, 09:57 PM #10 Newbie   Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 OK . . . . Finally, I think I have managed to convert the document into an appropriate format for the forum. Hopefully I can now move forward and get someone that knows more than me to help, answer some questions, and - just maybe - rekindle a love for math I used to have. So . . . Before I go ahead with my questions, can someone please check my working/calculations within the attachments? I went through them again manually over the weekend. But, since I rarely do this stuff by hand and/or without a software package these days, the chances that I have missed something out and/or gotten something wrong will probably not be insignificant. Cheers, Jim. Separable Partial Diff Task1 (Page 1).jpg Separable Partial Diff Task1 (Page 2).jpg Separable Partial Diff Task1 (Page 3).jpg

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