Differential Equations - Exact ODE's 
Hi all, I've attempted the question part b) after rearranging the 1st expression to :
$$v(x,y)\, dx - u(x,y) \, dy = 0.$$
After this I tried using the method of differentiating each expression and trying to calculate if they were exact. It was very lengthy so I am unable to post on here but it seems there is an easier way to go about this. Any help is appreciated.

$$(-v)dx+(u)dy=\frac{2xy}{(x^2+y^2)^2}dx+\left(1+\frac{y^2-x^2}{(x^2+y^2)^2} \right)dy=0$$
$$\frac{\partial (-v)}{\partial y}= \frac{\partial }{\partial y}\left(\frac{2xy}{(x^2+y^2)^2} \right)= \frac{2x}{(x^2+y^2)^2}-\frac{8xy^2}{(x^2+y^2)^3} =\frac{2x(x^2-3y^2)}{(x^2+y^2)^3}$$
$$\frac{\partial u}{\partial x}= \frac{\partial}{\partial x}\left(1+\frac{y^2-x^2}{(x^2+y^2)^2} \right)=\frac{-2x}{(x^2+y^2)^2}-\frac{4x(y^2-x^2)}{(x^2+y^2)^3} = \frac{2x(x^2-3y^2)}{(x^2+y^2)^3}$$
$$\frac{\partial (-v)}{\partial y}= \frac{\partial u}{\partial x}\quad\text{Thus the ODE is exact}$$
Solving the ODE :
$$\int(-v)dx=\int\frac{2xy}{(x^2+y^2)^2}dx=\frac{-y}{(x^2+y^2)}+f(y)$$
$$\int(u)dx=\int\left(1+\frac{y^2-x^2}{(x^2+y^2)^2}\right)dx=\frac{-y}{(x^2+y^2)}+y+g(x)$$
$$\frac{-y}{(x^2+y^2)}+f(y)=\frac{-y}{(x^2+y^2)}+y+g(x) \quad\implies\quad f(y)=y\text{ and } g(x)=0$$
The ODE is transformed into : $\quad d\left(\frac{-y}{(x^2+y^2)}+y\right)=0$. Integrating leads to the solution of the ODE :
$$\frac{-y}{(x^2+y^2)}+y=C$$
