Determine if the equation is exact. if it is$,$ then solve it$.$

$(3x^{2}y+3)dx+(x^3-5)dy=0$

The equation is exact and an implicit solution is $\frac{3}{2}{x}^2{y}^2+3y+\frac{x^4}{4}-5x=C,$ where $C$ is an arbitrary constant.

The equation is exact and an implicit solution is $x^{3}y+3x+x^{3}y-5y=C,$ where $C$ is an arbitrary constant.

The equation is not exact.

The equation is exact and an implicit solution is $x^{3}y+3x-5y=C,$ where $C$ is an arbitrary constant.

Solve the initial value problem.

$(2y{e}^{2xy}-\frac{7}{y})dx+(2x{e}^{2xy}+\frac{7x}{y^2})dy=0,y(1)=2$

The implicit solution is:

$e^{2xy}-\frac{7x}{y}=e^2+\frac{7}{2}$

$e^{2xy}-\frac{7x}{y}=e^2-\frac{7}{2}$

$e^{2xy}-\frac{7x}{y}=e^4+\frac{7}{2}$

$e^{2xy}-\frac{7x}{y}=e^4-\frac{7}{2}$