\( ( x ^ { 2 } + y ^ { 2 } ) d x + ( 2 x y ) d y = 0 \)

To solve the differential equation \( ( x ^ { 2 } + y ^ { 2 } ) d x + ( 2 x y ) d y = 0 \), we can rearrange it into the standard form of a differential equation: \[ \frac{dy}{dx} = -\frac{x^2 + y^2}{2xy} \] This transformed form suggests that the solution can be approached through techniques involving separable variables or homogeneous functions, as both sides have variables \( x \) and \( y \). Another effective method is using substitution, specifically polar coordinates \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). In such cases, the differential equation can take on simpler forms which are easier to manage and can lead to general or particular solutions swiftly! Another note: Keep an eye out for common mistakes! A frequent pitfall is neglecting to check if the equation satisfies the condition for exactness before assuming a straightforward integration. It pays to double-check that your rearrangements hold true at every step!