January 17th, 2015, 06:35 PM  #1 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36  Differential Equation.
Solve the differential equation. $\displaystyle (2xy^2y)dx+(y^2+x+y)dy=0$ 
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 nonzero $y$, you can divide by $y^2$to make the equation exact, and then integrate. The equation permits piecewisedefined solutions. Don't assume that y is positive.

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 
$\displaystyle (2xy)dx+(x+y)dy=0$ 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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