
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
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 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, 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 $\displaystyle 1/( x^2+ 1)$? 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. 
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  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,179 Thanks: 391 Math Focus: Yet to find out.  Quote:
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 

Tags 
differential, equations, order, problems, separation, variables 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Solve differential equation with use of separation of variables  MathAboveMeth  Differential Equations  10  December 22nd, 2016 04:09 AM 
Differential Equations (Separation of Variable)  MatureAgeStudent  Calculus  4  May 16th, 2015 06:28 PM 
2nd order Differential equations  jakeward123  Differential Equations  6  March 17th, 2011 03:23 PM 
first order differential equations  riotsandravess  Differential Equations  4  December 6th, 2010 09:20 AM 
differential equations of first order  diegosened  Differential Equations  1  January 1st, 1970 12:00 AM 