I have the following equation

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

and I am told it is separable, but not knowing how that is, I went ahead and solved it using the Exact method.

Let$M=x{y}^{2}+x\text{}\text{and}\text{}N=y{x}^{2}+y$

$My=2xy\text{}\text{and}\text{}Nx=2xy$

$\int M.dx\Rightarrow \int x{y}^{2}+x={x}^{2}{y}^{2}+\frac{{x}^{2}}{2}+g\left(y\right)$

Partial of$({x}^{2}{y}^{2}+\frac{{x}^{2}}{2}+g\left(y\right))\Rightarrow x{y}^{2}+{g\left(y\right)}^{\prime}$

${g\left(y\right)}^{\prime}=y$

$g\left(y\right)=\frac{{y}^{2}}{2}$

the general solution then is

$C={x}^{2}\frac{{y}^{2}}{2}+\frac{{x}^{2}}{2}+\frac{{y}^{2}}{2}$

Is this solution the same I would get if I had taken the Separate Equations route?

and I am told it is separable, but not knowing how that is, I went ahead and solved it using the Exact method.

Let

Partial of

the general solution then is

Is this solution the same I would get if I had taken the Separate Equations route?