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January 24th, 2015, 06:28 AM  #1 
Newbie Joined: Jan 2015 From: Minnesota Posts: 2 Thanks: 0  Simplifying in diff eq
So issue I am having involves the diff eq x * dy/dx = y + √(x²  y²) In my book it says to first divide both sides by x to get dy/dx = (y/x) + √(1  (y/x)²) For the life of me, I do not understand how the first equation simplifies to the second one. Am I missing some fundamental fact about powers? Any help would be greatly appreciated. Last edited by skipjack; January 24th, 2015 at 07:48 AM. 
January 24th, 2015, 06:46 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
$\displaystyle \frac{\sqrt{x^2y^2}}{x}=\frac{\sqrt{x^2y^2}}{ \sqrt{x^2}}=\sqrt{\frac{x^2y^2}{x^2}}=\sqrt{1 \left(\frac{y}{x}\right)^2}$

January 24th, 2015, 07:44 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,809 Thanks: 2150 
Does the book suggest one then makes a substitution? The substitution y = xu could have been done at the outset. It's best to consider x < 0 and x > 0 separately. What happens as x approaches zero is "interesting".

January 24th, 2015, 08:21 AM  #4 
Newbie Joined: Jan 2015 From: Minnesota Posts: 2 Thanks: 0  Sweet
Ah. Thank you. I knew I just wasn't seeing it the right way. Yes that is the sub the book makes.


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