
Differential Equations Ordinary and Partial Differential Equations Math Forum 
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April 7th, 2017, 08:36 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 8th, 2017 at 12:24 AM. 
April 8th, 2017, 12:29 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,250 Thanks: 1439 
Welcome to the forum. We'll try to help you if you post your questions.

April 10th, 2017, 02:48 PM  #3 
Newbie Joined: Apr 2017 From: New Zealand Posts: 7 Thanks: 0 
OK, thanks SJ. I need to make sure I have rewritten 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, 04:37 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 $\displaystyle 1/( x^2+ 1)$? Cheers, Jim. Last edited by skipjack; April 10th, 2017 at 08:47 PM. 
April 10th, 2017, 08:25 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 18,250 Thanks: 1439 
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. 
April 11th, 2017, 04:21 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 14th, 2017, 12:12 AM  #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 14th, 2017, 12:26 AM  #8  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,460 Thanks: 489 Math Focus: Yet to find out.  Quote:
Also, you can start a new thread when you do.  
April 14th, 2017, 12:46 AM  #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, 10:57 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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differential, equations, order, problems, separation, variables 
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