Why is the term -xy included only once in the final solution?

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.

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.

To check if a differential equation is exact, you must verify that the partial derivative of the coefficient of dx with respect to y is equal to the partial derivative of the coefficient of dy with respect to x. If these mixed partial derivatives are equal, the equation is exact.

Conditions: The equation must be in the form $M(x,y)dx + N(x,y)dy = 0$.; M and N must have continuous first-order partial derivatives.

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.