Part CCSIR NET December 2024the-sign-of-the-discriminant-is-what-is-preserved

The sign of the discriminant is what is preserved

Consider the partial differential equation $4 (\partial^{2} u / \partial x^{2}) + 6 (\partial^{2} u / \partial x \partial y) + 3 (\partial^{2} u / \partial y^{2}) = 0$ . Which of the following partial differential equations can be obtained by a change of independent variables given by $(ξ, η)$ ᵗ = A(x, y)ᵗ for some 2 × 2 invertible matrix A with real entries?

A. $\partial^{2} w / \partial ξ^{2} + \partial^{2} w / \partial ξ \partial η + \partial^{2} w / \partial η^{2} = 0$ ✓
B. $\partial^{2} w / \partial ξ^{2} + \partial^{2} w / \partial η^{2} = 0$ ✓
C. $\partial^{2} w / \partial ξ^{2} + 4 (\partial^{2} w / \partial ξ \partial η) + \partial^{2} w / \partial η^{2} = 0$
D. $2 (\partial^{2} w / \partial ξ^{2}) + 2 (\partial^{2} w / \partial ξ \partial η) + \partial^{2} w / \partial η^{2} = 0$ ✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: A PDE has a single type throughout its domain

The sign of the discriminant is what is preserved

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