Part BCSIR NET December 2024the-euler-operator-means-homogeneous

The euler operator means homogeneous

Let u = u(x, y) be the solution of the Cauchy problem $x \partial{}u/\partial{}x + y \partial{}u/\partial{}y = u, (x, y) \ne (0, 0)$ , with $u(x, 1) = \sqrt{1 + x^{2}}, x \in \mathbb{R}$ . Then which of the following statements is true?

A.u(1, 0) = 0
B. $u(x_{1}, y_{1}) = u(x_{2}, y_{2})$ whenever $x_{1}^{2} + y_{1}^{2} = x_{2}^{2} + y_{2}^{2}$ ✓ Correct
C. $u(1, y) = \sqrt{2}$ for all $y \in \mathbb{R}$
D. $u(x_{1}, y_{1}) = u(x_{2}, y_{2})$ whenever $x_{1} + y_{1} = x_{2} + y_{2}$

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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The euler operator means homogeneous

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