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 July 18th, 2013, 10:11 AM #11 Senior Member   Joined: Sep 2012 Posts: 112 Thanks: 0 Re: 2 exact differential equation Thank you so much Mark! My Q.1 has been solved too.
 July 19th, 2013, 10:53 AM #12 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: 2 exact differential equation Another method, cleverly suggested here, is to begin with: $$$\(x^2+y^2$$x-y\)\,dx+$$\(x^2+y^2$$y+x\)\,dy=0$ Now, switching to polar coordinates, we then eventually get: $r\,dr+d\theta=0$ Integrating, we obtain: $r^2+2\theta=C$ Back substituting for $r$ and $\theta$, we get: $x^2+y^2+2\tan^{\small{-1}}$$\frac{y}{x}$$=C$ Using the identity: $\tan^{\small{-1}}$$\frac{y}{x}$$+\tan^{\small{-1}}$$\frac{x}{y}$$=\frac{\pi}{2}$ we see this is equivalent to the result I gave before.
 July 19th, 2013, 09:21 PM #13 Senior Member   Joined: Sep 2012 Posts: 112 Thanks: 0 Re: 2 exact differential equation Thank you so much! To me the earlier solution is easier.

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