 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/20/a/ii/solution

x X^{'} Y - 2 y X Y^{'} = 0

gives

\frac{x X ^{'}}{X} = \frac{2 y Y ^{'}}{Y} = λ .

x, y > 0

, integration yields

X = A x^{λ}

and

Y = B y^{λ /2}

λ = 2

A B = 1

gives a polynomial that extends smoothly to all real

x, y

u (x, y) = x^{2} y .

It is nonconstant and real everywhere, and

x u_{x} - 2 y u_{y} = 2 x^{2} y - 2 x^{2} y = 0

, including on the axes.

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