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January 24th, 2015, 06:28 AM   #1
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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.
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January 24th, 2015, 06:46 AM   #2
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$\displaystyle \frac{\sqrt{x^2-y^2}}{|x|}=\frac{\sqrt{x^2-y^2}}{ \sqrt{x^2}}=\sqrt{\frac{x^2-y^2}{x^2}}=\sqrt{1- \left(\frac{y}{x}\right)^2}$
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January 24th, 2015, 07:44 AM   #3
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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".
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January 24th, 2015, 08:21 AM   #4
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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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