Checking Exactness for Differential Equation

Check for exactness: $2xy \, dx + x^2 \, dy = 0$.

Determine if this equation is exact by finding if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$

In this problem, you are asked to determine if a given differential equation is exact. Exact differential equations are an important class of equations in the study of differential equations. They often appear in contexts where the relationship between two variables can be expressed as a total differential. To check for exactness, you must determine if there exists a potential function whose total differential matches the given equation. The problem provides a specific test for exactness, which is verifying that the partial derivative of the first function with respect to y equals the partial derivative of the second function with respect to x.

The process of checking for exactness involves comparing two partial derivatives. If the condition is met, it implies that there exists a potential function, often denoted as psi or phi, which satisfies the given total differential equation. This connection to a potential function is particularly valuable as it implies the existence of a conserved quantity or a first integral of the system, which can simplify both the understanding and solving of the differential equation system more generally. Understanding exactness is fundamental for solving differential equations efficiently. It can sometimes lead to simplified solutions and can help in identifying integrating factors if the equation is not initially exact. This kind of problem typically appears in courses dealing with first-order differential equations, as it builds a foundation for more complex methods used in higher-order equations and systems.