\( y^{\prime }=(x+y+1)^{2} \)

This differential equation has a charming historical background, tracing its roots to early explorations in calculus. Mathematicians like Leibniz and Newton laid the groundwork by studying functions and their rates of change, leading to the development of ordinary differential equations. Such equations became crucial in understanding everything from motion to population dynamics, helping set the stage for modern mathematical applications. To tackle an equation like this, a common approach would involve finding an integrating factor or using substitution methods. However, a common mistake is overlooking how to properly expand and simplify terms. Careful manipulation of the right-side polynomial can make solving for \( y \) easier, so always double-check your arithmetic to ensure you're on the right track!