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

This equation is a first-order differential equation that can be solved using the method of exact equations or through integrating factors. Rearranging the terms, we can recognize that it may be useful to check whether it is exact by computing the partial derivatives of the functions involved. An integration may lead us to a function \(F(x, y)\) whose differential equals zero on the left-hand side, giving us a path to the solution. Another common strategy is to manipulate and separate variables, as certain forms of similar equations lead to simpler integrations. Make sure to look out for potential mistakes like misapplying the product rule or overlooking constant factors when integrating. Wouldn't it be a bummer to find a small error after all that hard work?