How do you check if a differential equation is exact?

The final implicit solution for the example equation (-3x^2 - y)$dx + (3y^2 - x)dy = 0$ is -x^$3 + y^3$ - xy = C.

Conditions: The equation is exact.; The terms are combined correctly without double-counting.

The term -xy appears in both the integration of M with respect to x and the integration of N with respect to y. When combining the results to form the potential function, shared terms should only be included once to avoid double-counting.

Conditions: The term -xy is present in both antiderivatives.

When integrating a multivariable function with respect to one variable, the other variables are treated as constants. Therefore, the 'constant' of integration can be any function that depends only on the other variables, which in this case is y.

Conditions: The integration is with respect to x.; The function M depends on both x and y.

The presenter leaves the answer in implicit form because the equation contains multiple powers and products of y, making it cumbersome to solve explicitly for y.

Conditions: The solution contains multiple y terms.; An explicit solution for y is not attempted.

The steps involve finding the potential function $F(x,y)$ by integrating M with respect to x and N with respect to y, then combining the results to form the implicit solution $F(x,y) = C$.

Conditions: The differential equation has been verified as exact.