Question

Show that if u(x, y) and v(x, y) are integrating factors of M(x, y) *dx + N(x, y) * dy = 0. (A) such that u(x, y) / v(x, y) is not constant, then u(x, y) = cv(x, y) is a solution of Equation (A) for every constant c.




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Answer 1

Answer:

Imagine we do these further partial derivatives:

∂M∂y = ∂2I∂y ∂x

∂N∂x = ∂2I∂y ∂x

they end up the same! And so this will be true:

∂M∂y = ∂N∂x

When it is true we have an an "exact equation" and we can proceed.

And to discover I(x, y) we do EITHER:

I(x, y) = ∫M(x, y) dx (with x as an independent variable), OR

I(x, y) = ∫N(x, y) dy (with y as an independent variable)

And then there is some extra work (we will show you) to arrive at the general solution

I(x, y) = C

Step-by-step explanation:

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