Final answer:
A separable differential equation M(x) + N(y)y' = 0 is exact because the partial derivatives of M with respect to y and N with respect to x are both zero, satisfying the condition for exactness.
Step-by-step explanation:
To show that a separable differential equation of the form M(x) + N(y)y' = 0 is also exact, we need to understand what it means for an equation to be exact. An exact equation is one where there exists a function Ψ(x, y) such that the total differential dΨ = M(x)dx + N(y)dy. In our case, M(x) depends only on x and N(y) depends only on y. Taking the partial derivative of M with respect to y and N with respect to x, we would get zero in both cases since M does not depend on y and N does not depend on x. Hence, the condition for exactness is satisfied trivially as:
∂M/∂y = 0 (since M is a function of x only)
∂N/∂x = 0 (since N is a function of y only)
Therefore, the separable equation is also exact, and there must exist a potential function Ψ(x, y) such that Ψ_x = M(x) and Ψ_y = N(y), which confirms the exactness of the given equation.