$(3)($i$)$ Show that the following equation is exact:

$2xy-9x^2+(2y+x^2+1)y^{\text{'}}=0.$

$($ii$)$ Find the general solution of this equation.

Sln: $($i$)M_y=N_x=2x,$ where ubrace$((2xy-9x^2)ubrace{)}_{M}dx+$ubrace$((2y+x^2+1)ubrace{)}_{N}dy=0$

$($ii$)$ From the system $(f_x,f_y)=(M,N),$ we first take integral from $M$ respect to $x,$ then take derivative of the result with respect to $y$ and compare it with $N,$ we get the general $($implicit$)$ solution $y^2+(x^2+1)y-3x^3=c$ after taking integral respect to $y.$