(e^y+ye^x) dx+ (xe^y+e^x) dy=0 what is the solution?

Question

diferantial equation

Answer 1

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=...