Test for exactness:

\( \frac{ \partial M}{ \partial x } = 2\) and \( \frac{ \partial N}{ \partial y } = 2\), thus exact equation

\( \frac{ \partial F}{ \partial x} = M = x + 2y\)

\( \frac{ \partial F}{ \partial y} = N = 2x + y \)

\( F = \frac{x^2}{2} +2xy + p(y) \)

\( \frac{\partial F}{ \partial y } = 2x + p'(y) \)

\( \frac{\partial F}{ \partial y } = 2x + p'(y) = 2x + y\)

\( p'(y) = y\)

\( p(y) = \frac{y^2}{2} \)

\(F = \frac{x^2}{2} +2xy + \frac{y^2}{2} \)

\( c = x^2 + 4xy + y^2 \)