Differential Equation

**Oblivionwarrior** · October 5th 2008, 02:58 PM

Hey guys, I am wondering if anyone can help me with this question.

Solve the given differential equation $\frac{dy}{dx} = -\frac{2xy+y^2+2}{x^2+2xy}$

Now the methods of solving I have learned so far are, using an integrating factor, separating the equation and the technique for exact equations using partial derivatives. Thanks!

**Chris L T521** · October 5th 2008, 03:36 PM

Originally Posted by Oblivionwarrior

Hey guys, I am wondering if anyone can help me with this question.

Solve the given differential equation $\frac{dy}{dx} = -\frac{2xy+y^2+2}{x^2+2xy}$

Now the methods of solving I have learned so far are, using an integrating factor, separating the equation and the technique for exact equations using partial derivatives. Thanks!

Note that $\frac{\,dy}{\,dx}=-\frac{2xy+y^2+2}{x^2+2xy}\implies (x^2+2xy)\,dy=-(2xy+y^2+2)\,dx$ $\implies (2xy+y^2+2)\,dx+(x^2+2xy)\,dy=0$

Can you try to take it from here? It should somewhat be obvious now...

--Chris

**Oblivionwarrior** · October 5th 2008, 03:58 PM

Originally Posted by Chris L T521

Note that $\frac{\,dy}{\,dx}=-\frac{2xy+y^2+2}{x^2+2xy}\implies (x^2+2xy)\,dy=-(2xy+y^2+2)\,dx$ $\implies (2xy+y^2+2)\,dx+(x^2+2xy)\,dy=0$

Can you try to take it from here? It should somewhat be obvious now...

--Chris

Yea I got to that point, but the equation isn't exact since the partial derivatives are not equal. So I need to find an integrating factor. (Unless I did the partial derivatives wrong and it is exact).

**Chris L T521** · October 5th 2008, 04:03 PM

Originally Posted by Oblivionwarrior

Yea I got to that point, but the equation isn't exact since the partial derivatives are not equal. So I need to find an integrating factor. (Unless I did the partial derivatives wrong and it is exact).

It is exact:

$M(x,y)=2xy+y^2+2$ and $N(x,y)=x^2+2xy$

Its exact when $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

$\frac{\partial M}{\partial y}=\color{red}2x+2y$

$\frac{\partial N}{\partial x}=\color{red}2x+2y$

Can you continue now?

--Chris

**Oblivionwarrior** · October 5th 2008, 04:25 PM

Wow it is exact, no wonder.

I thought M(x,y) = $x^2 +2xy$

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