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An exact differential equation is a type of first-order equation that can be expressed in the form M(x, y)dx + N(x, y)dy = 0, where M and N are functions of x and y. These equations can be solved by finding a function, often denoted as F(x, y), whose partial derivatives with respect to x and y are equal to M and N respectively. To classify an equation as exact, the partial derivative of M with respect to y must be equal to the partial derivative of N with respect to x.

In the above solution, this check confirms that the given differential equation is exact. This means there exists a function F(x, y) such that dF = Mdx + Ndy. By integrating M with respect to x and N with respect to y, you can find this F(x, y), which leads to the general solution of the equation.

When a given differential equation is not exact, you can sometimes make it exact by multiplying through by an integrating factor. An integrating factor is a function that depends on the variables x, y, or both, and it transforms a non-exact differential equation into an exact one.

The goal is to find a function μ(x, y) such that when multiplied by the non-exact equation, it yields an exact equation, that is, μMdx + μNdy = 0 becomes exact. Determining the appropriate integrating factor generally involves some trial and error and might rely on the specific forms of M and N. But in the problem described, the equation was already exact, so finding an integrating factor was unnecessary.

The general solution of a differential equation represents the family of all possible solutions that satisfy the equation. For exact differential equations, the general solution takes the form of F(x, y) = C, with C being an arbitrary constant. To arrive at this general solution, integration is performed with respect to x and y, following the confirmations that the equation is exact.

The standard approach often involves integrating M(x,y) while treating y as a constant and N(x,y) while treating x as a constant. The resulting functions are then combined, adjusting for any duplication of the mixed x, y terms, to form the complete solution.

Partial derivatives play a crucial role in the analysis of differential equations, especially when determining if an equation is exact. The partial derivative of a function with respect to one variable, say x, is its derivative with all other variables held constant.

In the context of exact differential equations, if M and N are continuous and have continuous first partial derivatives on a domain, then the equation is exact on that domain if ∂M/∂y = ∂N/∂x.

In the given problem, finding these partial derivatives was integral to confirming that the differential equation was exact, paving the way to finding its general solution. The step-by-step process followed in the solution, calculating ∂M/∂x and ∂N/∂y, highlights their equality and, thereby, the exactness of the equation.