Consider the following differential equation.

$(x-y^5+y^{2}sin(x))dx=(5x{y}^4+2ycos(x))dy$

Let $M=y^5-y^{2}sin(x)-x$ and $N=5x{y}^4+2ycos(x).$ Find the following partial derivatives.

$M_y=$

$N_x=$

$Is$ the given differential equation exact?

Yes

$No$

Let $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}=y^5-y^{2}sin(x)-x.$ Integrate this partial derivative with respect $tox,$ letting $h(y)bean$ unknown function $iny.$

$f(x,y)=,+h(y)$

Find the derivative $ofh(y).$

$h^{\text{'}}(y)=$

Find the general solution $of$ the given differential equation. $(Ifitis$ not exact, enter NOT.$)$