X φ y dy dx this means we are looking for equations

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∂x + ∂φ ∂y dy dx . This means we are looking for equations of the form M ( x, y ) + N ( x, y ) dy dx = 0 where there exists a function φ ( x, y ) such that M = ∂φ ∂x and N = ∂φ ∂y .
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From calculus, the following theorem gives us a very useful condition to know when such a function φ exists: Theorem Let M ( x, y ) and N ( x, y ) be continuous with continuous partial derivatives in some region R that is the interior of a simple closed curve. Then there exists a function φ ( x, y ) such that M = ∂φ ∂x and N = ∂φ ∂y if and only if ∂M ∂y = ∂N ∂x on the region R . I Example Consider the equation ye xy + (2 y + xe xy ) dy dx = 0 which is neither separable, nor linear. In this case we have M ( x, y ) = ye xy and N ( x, y ) = 2 y + xe xy . Computing partials gives ∂M
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