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To rewrite the equation in separable form, first move the term containing dx to the right-hand side to get 2ydy = ($1+\cos x$)($1+y^2$)dx. Then, divide both sides by ($1+y^2$) to isolate all terms involving y and dy on the left, and all terms involving x and dx on the right.

Conditions: The equation is a first-order differential equation.; The term ($1+y^2$) is never zero for real y, so division is always valid.

To rewrite the equation in separable form, first move the term containing dx to the right-hand side to get 2ydy = ($1+\cos x$)($1+y^2$)dx. Then, divide both sides by ($1+y^2$) to isolate all terms involving y and dy on the left, and all terms involving x and dx on the right.

Conditions: The equation is a first-order differential equation.; The term ($1+y^2$) is never zero for real y, so division is always valid.