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January 17th, 2015, 06:35 PM   #1
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Differential Equation.

Solve the differential equation.

$\displaystyle (2xy^2-y)dx+(y^2+x+y)dy=0$
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January 17th, 2015, 07:42 PM   #2
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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.
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January 17th, 2015, 07:44 PM   #3
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So this is not a differential equation?
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January 17th, 2015, 07:58 PM   #4
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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.
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January 17th, 2015, 08:06 PM   #5
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How to solve it then ? Can anyone guide me ?
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January 17th, 2015, 08:50 PM   #6
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How far can you get by using what I suggested above?
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January 18th, 2015, 02:16 AM   #7
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Do you mean that divide the equation by y^2?
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January 18th, 2015, 04:42 AM   #8
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Yes.
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January 18th, 2015, 10:59 PM   #9
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$\displaystyle (2x-y)dx+(x+y)dy=0$

How to continue?
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January 19th, 2015, 03:06 AM   #10
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Substitute y = xu. The resulting equation in u and x is separable.
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