91Ó°ÊÓ

Partial derivatives measure how a function changes as its variables change, one at a time. In a multivariable function like \( M(x, y) = 4x - y \) and \( N(x, y) = 6y - x \), partial derivatives help us explore how each component behaves individually. For example, the partial derivative of \( M \) with respect to \( y \) is calculated by treating \( x \) as a constant and only differentiating with respect to \( y \). In this case, we get \( \frac{\partial M}{\partial y} = -1 \). Similarly, the partial derivative of \( N \) with respect to \( x \) is calculated by considering \( y \) as constant, resulting in \( \frac{\partial N}{\partial x} = -1 \). By comparing these cross partial derivatives, we can determine if the differential equation is exact.

Integration is a fundamental concept in calculus used to find functions given their derivatives. In the context of exact differential equations, integrating the function \( M(x, y) \) with respect to \( x \) while treating \( y \) as constant helps us progress toward the solution. For the function \( M(x, y) = 4x - y \), integrating with respect to \( x \) gives \( \int (4x - y) \, dx = 2x^2 - yx + g(y) \), where \( g(y) \) remains an arbitrary function of \( y \). This step is crucial in forming the implicit solution as it begins combining components influenced by both variables. The integration process, therefore, transforms the differential form back into a potential F function, indicating total differentials.

In differential equations, an implicit solution is often encountered, where the solution is not distinctly solved for one variable. The equation \( 2x^2 - yx + 3y^2 = C \) serves as an implicit solution to the given problem. Rather than rewriting \( y \) explicitly as a function of \( x \), or vice versa, this formulation leaves \( x \) and \( y \) intertwined in an equation. Such solutions are particularly useful when isolating a single variable would be cumbersome. Implicit solutions suggest a set of curves, rather than a single function, where each curve satisfies the differential equation for different values of the integration constant \( C \). This method allows for a broader understanding of how solutions behave.

The exactness condition is a guiding principle in determining whether a differential equation can be solved as an exact equation. For a differential equation of the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), exactness requires that the partial derivative of \( M \) with respect to \( y \) must equal the partial derivative of \( N \) with respect to \( x \). In our example, these conditions hold as both \( \frac{\partial M}{\partial y} = -1 \) and \( \frac{\partial N}{\partial x} = -1 \). This equality indicates that the original equation represents the total differential of some potential function \( F(x, y) \), meaning it can be integrated directly to find a solution. This property simplifies the problem by confirming that methods for exact equations are applicable.