Question

Determine whether the equation is exact. If it is, then solve it. (- sin x sin y-8x)dx + (COS X cos y - 10y)dy = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer 1

The equation is exact and can be solved by finding a function φ(x, y) that satisfies the partial derivatives. The general solution is
`φ(x, y) = -x^2sin(y) - 4x^2 + sin(x)cos(y) - 5y^2 + h(x)`

Step-by-step explanation:

The equation is not exact because the mixed partial derivatives of the coefficients do not match. To determine whether an equation is exact, we check if
`∂M/∂y = ∂N/∂x` N are the coefficients of dx and dy respectively. In this case,
`∂M/∂y = -cos(x)sin(y) and ∂N/∂x = -cos(x)sin(y)` derivatives are equal.

To solve the equation, we need to find a function φ(x, y) such that
`∂φ/∂x = (-sin(x)sin(y) - 8x) and ∂φ/∂y = (cos(x)cos(y) - 10y)`
`(y) - 4x^2 + g(y)` an arbitrary function of y.

Taking the partial derivative of φ(x, y) with respect to y and comparing it to ∂φ/∂y, we find g'(y) = cos(x)cos(y) - 10y. Integrating g'(y) with respect to y gives
`g(y) = sin(x)cos(y) - 5y^2 + h(x)` function of x. Therefore, the general solution of the equation is
`φ(x, y) = -x^2sin(y) - 4x^2 + sin(x)cos(y) - 5y^2 + h(x)`.

Answer 2

Answer: Mabel is d

Step-by-step explanation: