<s>[INST] Solve the ODE $\cot (x^2+y^2)dy+xdx+ydy=0$ Solve the ODE $\cot (x^2+y^2)dy+xdx+ydy=0$
i am trying to solving integrating combination
since given that 
$\cot (x^2+y^2)dy+xdx+ydy=0$
then $\cot (x^2+y^2)dy+d(xy)=0$ is it correct way ? and we can apply integration from here? can any one help me this problem
 [/INST] Dividing both sides of $\cot (x^2+y^2)dy+xdx+ydy=0$ by $dy$ we get $\cot (x^2+y^2)+x\frac {dx}{dy}+y=0$ 
Then, 
$$x\frac {dx}{dy}=-y-\cot (x^2+y^2) \iff \frac {dx}{dy}=\frac{-y-\cot (x^2+y^2)}{x} \iff \frac {dy}{dx}=-\frac{x}{y+\cot (x^2+y^2)}$$
Now you can integrate this: $\frac {dy}{dx}=-\frac{x}{y+\cot (x^2+y^2)}$
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