**Unformatted text preview: **xy ) x ] = 2 x cos( xy ) − x 2 y sin( xy ) , ∂N ∂x = 0 + [2 x cos( xy ) − x 2 sin( xy ) y ] = 2 x cos( xy ) − x 2 y sin( xy ) . Therefore, the equation is exact. So, we use the method discussed in Section 2.4 and obtain F ( x, y ) = Z N ( x, y ) dy = Z ± 1 + x 2 cos( xy ) ² dy = y + x sin( xy ) + h ( x ) ⇒ ∂F ∂x = sin( xy ) + x cos( xy ) y + h ( x ) = M ( x, y ) = sin( xy ) + xy cos( xy ) ⇒ h ( x ) = 0 ⇒ h ( x ) ≡ , and a general solution is given implicitly by y + x sin( xy ) = c . 91...

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