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 January 17th, 2015, 06:35 PM #1 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 Differential Equation. Solve the differential equation. $\displaystyle (2xy^2-y)dx+(y^2+x+y)dy=0$
 January 17th, 2015, 07:42 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 Math Focus: Mainly analysis and algebra If the $-y$ were $+y$ or the $+x$ were $-x$ I think we could get an integrating factor (in $y$) to make it an exact differential equation.
 January 17th, 2015, 07:44 PM #3 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 So this is not a differential equation?
 January 17th, 2015, 07:58 PM #4 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 The equation is satisfied by $y = 0$. For non-zero $y$, you can divide by $y^2$to make the equation exact, and then integrate. The equation permits piecewise-defined solutions. Don't assume that y is positive. Thanks from topsquark and szz
 January 17th, 2015, 08:06 PM #5 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 How to solve it then ? Can anyone guide me ?
 January 17th, 2015, 08:50 PM #6 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 How far can you get by using what I suggested above?
 January 18th, 2015, 02:16 AM #7 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 Do you mean that divide the equation by y^2?
 January 18th, 2015, 04:42 AM #8 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 Yes.
 January 18th, 2015, 10:59 PM #9 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 $\displaystyle (2x-y)dx+(x+y)dy=0$ How to continue?
 January 19th, 2015, 03:06 AM #10 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 Substitute y = xu. The resulting equation in u and x is separable.

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