What to use when reverse chain rule doesnt work?

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SUMMARY

The discussion focuses on solving the differential equation (xy+(x^2))dx + (-1)dy=0, where the method of exact solutions fails due to unequal partial derivatives. The reverse chain rule is identified as an ineffective approach in this case. Instead, the equation can be transformed into an exact form by manipulating the expression to conclude that it is exact in the coordinates derived from the transformation. This allows for the integration of the modified equation to find a solution.

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highc1157
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Hi there,

My equation to solve is (xy+(x^2))dx + (-1)dy=0

For method of exact solutions, the partials are not equal to each other so I cannot use
exact solutions (reverse chain rule)

I don't know how to solve this
 
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What is reverse chain rule?
Any equation is exact in some coordinates so find them.
you stated (correctly)
(xy+x2)y - (-1)x
is not zero but we can use the fact that
(((xy+x2)y -(-1)x)/(-1))y=0
to conclude that the equation is exact in
∫((xy+x2)y - (-1)x)/(-1)∂x
and y since (xy+(x2))dx + (-1)dy can be written
(-y-sqrt(-2(-x2/2)))d(-x2/2)-dy
 

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