 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/14/c/solution

Let

t = y / x

u_{x} = ϕ (t) + x ϕ^{'} (t) (- \frac{y}{x ^{2}}) = ϕ (t) - t ϕ^{'} (t), u_{y} = x ϕ^{'} (t) \frac{1}{x} = ϕ^{'} (t) .

Hence the terms containing

ϕ^{'}

cancel:

x u_{x} + y u_{y} = x ϕ (t) - x t ϕ^{'} (t) + y ϕ^{'} (t) = x ϕ (t) = u .

u (λ x, λ y) = λ u (x, y)

for

λ > 0

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