Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 1 / c / Solution

Use the change of variables $u=x+y$. Then $u^{\prime }=1+y^{\prime }=1+\cos u=2{\cos }^2(u/2)$, a separable differential equation. Where $u$ is not an odd multiple of $\pi$, separation of variables givesIndeed the derivative of $\tan (u/2)$ with respect to $u$ is $1/(1+\cos u)$. The nonconstant branches can also be written $y=2\arctan (x+C)+2k\pi -x$, $k\in \mathbb{Z}$. Dividing by $1+\cos u$ excluded its zeros: the missing equilibrium solutions for $u$ give the additional solutions of a differential equationThey satisfy $y^{\prime }=-1=\cos (x+y)$ directly. The transformed right side is continuously differentiable, so the uniqueness theorem for ordinary differential equations prevents a nonconstant branch from crossing an equilibrium solution at finite $x$.