Final answer:
To find the value of k so that the given differential equation is exact, we set the partial derivatives of M and N equal to each other and solve for k. The value of k is found to be 3/2.
Step-by-step explanation:
In order for the given differential equation to be exact, it must satisfy the condition of having equal partial derivatives in the correct form. Let's denote the given equation as M(x, y)dx + N(x, y)dy = 0. The condition for exactness is that ∂M/∂y = ∂N/∂x.
In this case, we have M(x, y) = 2xy³ + cos(y) and N(x, y) = 2kx²y² - x*sin(y).
By taking the partial derivatives and equating them, we get:
∂M/∂y = 6xy² - sin(y) and ∂N/∂x = 4kxy² - sin(y).
Setting these two expressions equal to each other, we have:
6xy² - sin(y) = 4kxy² - sin(y).
Cancelling out the sin(y) terms, we are left with:
6xy² = 4kxy².
Dividing both sides by xy², we get:
6 = 4k.
Finally, we solve for k:
k = 6/4 = 3/2.