(xy^2 + x - 2y + 3) dx + x^2 ydy = 2(x+y) dy
$(xy^2 + x - 2y + 3)\,dx + x^2y\,dy = 2(x + y)\,dy$
$(xy^2 + x - 2y + 3)\,dx + (x^2y - 2x - 2y)\,dy = 0$
$\dfrac{\partial M}{\partial y} = 2xy - 2$
$N = x^2y - 2x - 2y$
$\dfrac{\partial N}{\partial x} = 2xy - 2$
Hence, the given is an exact equation
$\partial F = M \, \partial x$
$F = (xy^2 + x - 2y + 3)\,\partial x$
$F = \frac{1}{2}x^2y^2 + \frac{1}{2}x^2 - 2xy + 3x + f(y)$
$\dfrac{\partial F}{\partial y} = x^2y - 2x + f'(y)$
$\dfrac{\partial F}{\partial y} = N$
$x^2y - 2x + f'(y) = x^2y - 2x - 2y$
$f'(y) = -2y$
$f(y) = -y^2$
Thus, $F = \frac{1}{2}x^2y^2 + \frac{1}{2}x^2 - 2xy + 3x - y^2$
$F = c$
$\frac{1}{2}x^2y^2 + \frac{1}{2}x^2 - 2xy + 3x - y^2 = c$ answer
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$(xy^2 + x - 2y + 3)\,dx + x^2y\,dy = 2(x + y)\,dy$
$(xy^2 + x - 2y + 3)\,dx + (x^2y - 2x - 2y)\,dy = 0$
$\dfrac{\partial M}{\partial y} = 2xy - 2$
$N = x^2y - 2x - 2y$
$\dfrac{\partial N}{\partial x} = 2xy - 2$
Hence, the given is an exact equation
$\partial F = M \, \partial x$
$F = (xy^2 + x - 2y + 3)\,\partial x$
$F = \frac{1}{2}x^2y^2 + \frac{1}{2}x^2 - 2xy + 3x + f(y)$
$\dfrac{\partial F}{\partial y} = x^2y - 2x + f'(y)$
$\dfrac{\partial F}{\partial y} = N$
$x^2y - 2x + f'(y) = x^2y - 2x - 2y$
$f'(y) = -2y$
$f(y) = -y^2$
Thus,
$F = \frac{1}{2}x^2y^2 + \frac{1}{2}x^2 - 2xy + 3x - y^2$
$F = c$
$\frac{1}{2}x^2y^2 + \frac{1}{2}x^2 - 2xy + 3x - y^2 = c$ answer
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