Final answer:
The function M(x, y) for the exact differential equation M(x, y) dx + (x² - y²) dy = 0 is M(x, y) = 2xy + h(x), with h(x) representing an arbitrary function of x.
Step-by-step explanation:
To find the function M(x, y) such that the differential equation M(x, y) dx + (x² - y²) dy = 0 is exact, we first need to understand the concept of an exact differential equation. This type of equation can be written in the form N(x, y) dy + M(x, y) dx = 0, where the function M is the partial derivative with respect to x of some potential function F(x, y), and N is the partial derivative with respect to y of F. For the equation to be exact, the condition ∂M/∂y = ∂N/∂x must be fulfilled.
In this case, we already have N(x, y) = x² - y², so we compute the partial derivative of N with respect to x, which is 2x. We then assume ∂M/∂y equals this derivative (2x) and integrate 2x with respect to y to find M(x, y). As the integration with respect to y does not affect x, x can be treated as a constant during integration. Thus, we obtain M(x, y) = 2xy + h(x), where h(x) is a function of x alone representing the constant of integration.
To determine the function h(x), we need to satisfy the initial condition that ∂M/∂x = ∂N/∂y, but since there are no further constraints given on M(x, y), h(x) remains an arbitrary function of x. Therefore, the general solution for M(x, y) is M(x, y) = 2xy + h(x), with h(x) as the arbitrary function of integration.