Exactness of Differential Equation and General Solution

Check for exactness: $\displaystyle (2x+4y-1) \, dx + (4x+6y+1) \, dy = 0$. Determine if this equation is exact by finding if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ and find the general solution of the exact differential equation.

When determining if a differential equation is exact, you need to verify that the mixed partial derivatives are equal. In this context, you will express the given differential equation in the form $M(x, y)dx + N(x, y)dy = 0$, where $M(x, y)$ and $N(x, y)$ are functions of x and y. The condition for exactness is that the partial derivative of $M$ with respect to y must be equal to the partial derivative of $N$ with respect to x. Checking for exactness essentially means determining whether the differential equation may have derived from a single function known as a potential function. This is due to the concept from multivariable calculus where the path integral between two points is independent of the path taken if a vector field $F$ is conservative, meaning the field is the gradient of some scalar field. In the context of a differential equation, this analogous process involves finding if such an underlying function, or potential function, exists. Once it is established that a differential equation is exact, solving it generally involves finding a potential function whose gradient yields the original differential equation components. This involves integrating $M$ with respect to x and $N$ with respect to y, then combining these results while ensuring consistency to form the general solution. Understanding this process is crucial in courses covering differential equations because it emphasizes the connection between different fields of mathematics such as calculus and differential equations and provides a systematic method to solve a certain class of differential equations efficiently.