Suppose an airplane with fixed air speed $v_{0}$ makes a flight lasting $T$ seconds. Suppose the air has velocity $0 < a < v_{0}$ in the positive $y$ direction. Find that closed curve along which the plane should fly if this curve is to enclose the greatest area.
Let the curve be denoted by $C$ and the region it bounds be denoted by $R$. We wish to maximize the functional $$ A[C] = \iint_{R}d(x,y) .$$ Green's theorem provides a method of computing this $$\iint_{R}d(x,y) = \int_{C}xdy = \int_{0}^{T}x(t)y'(t) \, dt .$$
Let that integrand be denoted by $F$ and the next be denoted by $G$. A constraint is that the airspeed is fixed, so $\sqrt{x'^{2}+(y'-a)^{2}} = v_{0}$ and $$Tv_{0} = \int_{0}^{T}\sqrt{x'^{2}+(y'-a)^{2}} \, dt$$. Put $F^{*} = F-\lambda G$ where $\lambda$ is a constant (Lagrange Multiplier) Any solutions $x$ and $y$ must satisfy $$ \frac{d}{dt}F_{x'}^{*} = F_{x}^{*} $$ $$ \frac{d}{dt}F_{y'}^{*} = F_{y}^{*} .$$
We know that the final answer should be an ellipse but this results in a differential equation that we (nor Mathematica) can solve. So there must be an error somewhere. Either in our understanding of the question (the constraint $G$ could be wrong?) or our execution.
Thank you.