Final answer:
After computing the partial derivatives ∂m/∂y and ∂n/∂x for each pair of m(x, y) and n(x, y) given, it can be verified that none of the differential equations are exact as the necessary condition for exactness is not met in any case.
Step-by-step explanation:
To verify whether the given differential equation is exact, we need to check if the conditions for exactness are satisfied. An exact equation requires that the partial derivative of m with respect to y is equal to the partial derivative of n with respect to x. Let's determine the exactness for each pair m(x, y) and n(x, y).
- For the first pair, m(x, y) = -xy sin(x) + 2y cos(x), and n(x, y) = 2x cos(x), we compute the partial derivatives:
- ∂m/∂y = -x sin(x) + 2 cos(x)
- ∂n/∂x = -2x sin(x) + 2 cos(x)
- The partial derivatives are not equal, thus the DE is not exact.
Repeating this check for the other given pairs:
- For the second pair, where n(x, y) = -2x cos(x), ∂n/∂x = 2x sin(x) - 2 cos(x), and again, the DE is not exact.
- For the third and fourth pair, the procedure would be the same, but regardless of the sign before the 2x cos(x), the partial derivatives will not match, indicating non-exactness.
Hence, none of the provided differential equations are exact, satisfying the condition for exactness.