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July 8th, 2017, 11:29 AM  #1 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Constant C in differential equation
Hi guys, $$\int {}\frac{1}{ky}dy =\int dx \rightarrow \frac{1}{k}\ln \left  y \right =x+C$$ Why is C only on the right side and not on the left side? Thanks in advance 
July 8th, 2017, 11:33 AM  #2 
Senior Member Joined: Nov 2015 From: United States of America Posts: 162 Thanks: 21 Math Focus: Calculus and Physics 
You can put a 'c' on both sides if you want. C is an arbitrary constant. You can combine all constants to form one final constant. $$\int {}\frac{1}{ky}dy =\int dx \rightarrow \frac{1}{k}\ln \left  y \right  + C1 =x+C2$$ $$\rightarrow \frac{1}{k}\ln \left  y \right  =x+C2  C1$$ $$\rightarrow C2  C1 = C$$ $$\rightarrow \frac{1}{k}\ln \left  y \right  =x+C $$ Last edited by skipjack; July 8th, 2017 at 08:45 PM. 
July 10th, 2017, 04:18 AM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,397 Thanks: 479 Math Focus: Yet to find out. 
Might also be worth keeping in mind that the general solution of an $nth$ order differential equation will contain $n$ arbitrary constants.
Last edited by skipjack; July 10th, 2017 at 04:46 AM. 

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