
Differential Equations Ordinary and Partial Differential Equations Math Forum 
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November 8th, 2016, 09:04 AM  #1 
Newbie Joined: Nov 2016 From: London Posts: 3 Thanks: 0  First order differential equation
Hey all, I have a separable differential equation: Can someone please explain how the solution is: Thanks 
November 8th, 2016, 09:25 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 Math Focus: Mainly analysis and algebra 
Post your attempt first. There's nothing that should cause a differential equations student much trouble, especially if you know how to integrate by partial fractions (writing your constant of integration as $\ln c$) and how to use logarithmic identities to good effect. Perhaps the hardest part is removing the absolute value signs. 
November 8th, 2016, 11:09 AM  #3 
Newbie Joined: Nov 2016 From: London Posts: 3 Thanks: 0 
I get 1/2ln(2x+1)=ln(t/(t+1))+ln(C), but what should I do then?
Last edited by skipjack; November 8th, 2016 at 02:49 PM. Reason: to insert parentheses 
November 8th, 2016, 02:54 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Can you write that equation in the form ln(...) = ln(...)?

November 8th, 2016, 03:38 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 Math Focus: Mainly analysis and algebra 
You are looking for the terms in $t$ to be squared, so multiply by 2 and use properties of the logarithm.
Last edited by skipjack; November 8th, 2016 at 03:41 PM. 
November 8th, 2016, 03:42 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Mrkukas, are you given that t > 0?

November 8th, 2016, 04:42 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 Math Focus: Mainly analysis and algebra 
He shouldn't be. It should be a valid result for all $t$ with the possible exception of $t=1$.

November 8th, 2016, 06:16 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
That depends on what you mean by "valid result" and "possible exception". The original equation doesn't imply that t isn't 1, but the supplied solution does, as it's undefined at t = 1. 
November 8th, 2016, 11:34 PM  #9 
Newbie Joined: Nov 2016 From: London Posts: 3 Thanks: 0  Last edited by skipjack; November 8th, 2016 at 11:51 PM. 
November 8th, 2016, 11:53 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Can you assume that t is real?


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