Rewrite:
eʸ+yeˣ+xeʸdy/dx+eˣdy/dx=0.
Rearrange:
(eʸ+xeʸdy/dx)+(yeˣ+eˣdy/dx)=0.
This is just a double expansion of the product rule:
d(xeʸ)/dx+d(yeˣ)/dx=0.
This can sometimes be written:
Dᵪ(xeʸ)+Dᵪ(yeˣ)=0.
Integration simple gives us:
xeʸ+yeˣ=C, where C is the constant of integration.
This is an implicit integration because it can’t be written in the form y=...