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Show that any separable equation M(x) N(y)y′ = 0 is also exact.

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Final answer:

A separable equation of the form M(x) N(y)y' = 0 is exact because we can find a function F(x, y) satisfying the conditions that the partial derivatives with respect to x and y are M(x) and N(y), respectively.

Step-by-step explanation:

To show that any separable 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 separable and exact. A separable equation is one that can be written as a product of functions where each function exclusively contains one variable, in this case, M(x) and N(y). An exact equation, on the other hand, is one where there exists a function F(x, y) such that ∂F/∂x = M(x) and ∂F/∂y = N(y). In our case, since M(x) only depends on x and N(y) only depends on y, we can find a function F(x, y) = ∫M(x)dx + ∫N(y)dy which satisfies these conditions, thereby making our separable equation exact.

User Steven Kryskalla
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