Question

mostre que a EDO (4xy^2+4x+y)dx + (4x^2y+x4y)dy=0 é exata.
amigos alguém pode me explicar passo a passo como faço essa EDO.

Answer 1

Identifiquemos
             $M=4xy^2+4x+y\to \boxed{M_y=8xy+1} \\ \\ N=4x^2y+x+4y\to \boxed{N_x=8xy+1}$

Puesto que $M_y=N_x$ la EDO es exacta.

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         $M=4xy^2+4x+y\\ \\ M_y=\dfrac{\partial}{\partial y}(4xy^2+4x+y)\\ \\ M_y=\dfrac{\partial}{\partial y}(4xy^2)+\dfrac{\partial}{\partial y}(4x)+\dfrac{\partial}{\partial y}(y)\\ \\ M_y=4x\dfrac{\partial}{\partial y}(y^2)+\dfrac{\partial}{\partial y}(4x)+\dfrac{\partial}{\partial y}(y)\\ \\ M_y=4x(2y)+0+1\\ \\ \boxed{M_y=8xy+1} \\ \\ \\.$


        $N=4x^2y+x+4y\\ \\ N_x=\dfrac{\partial}{\partial x}(4x^2y+x+4y)\\ \\ N_x=\dfrac{\partial}{\partial x}(4x^2y)+\dfrac{\partial}{\partial x}(x)+\dfrac{\partial}{\partial x}(4y)\\ \\ N_x=4y\dfrac{\partial}{\partial x}(x^2)+\dfrac{\partial}{\partial x}(x)+\dfrac{\partial}{\partial x}(4y)\\ \\ N_x=4y(2x)+1+0\\ \\ \boxed{N_x=8xy+1}$