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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(∂2u/∂x2)+6(∂2u/∂x∂y)+3(∂2u/∂y2)=04(\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)(\xi, \eta)ᵗ = A(x, y)ᵗ for some 2 × 2 invertible matrix A with real entries?

  1. A.∂2w/∂ξ2+∂2w/∂ξ∂η+∂2w/∂η2=0\partial^{2}w/\partial\xi^{2} + \partial^{2}w/\partial\xi\partial\eta + \partial^{2}w/\partial\eta^{2} = 0✓ Correct
  2. B.∂2w/∂ξ2+∂2w/∂η2=0\partial^{2}w/\partial\xi^{2} + \partial^{2}w/\partial\eta^{2} = 0✓ Correct
  3. C.∂2w/∂ξ2+4(∂2w/∂ξ∂η)+∂2w/∂η2=0\partial^{2}w/\partial\xi^{2} + 4(\partial^{2}w/\partial\xi\partial\eta) + \partial^{2}w/\partial\eta^{2} = 0
  4. D.2(∂2w/∂ξ2)+2(∂2w/∂ξ∂η)+∂2w/∂η2=02(\partial^{2}w/\partial\xi^{2}) + 2(\partial^{2}w/\partial\xi\partial\eta) + \partial^{2}w/\partial\eta^{2} = 0✓ Correct

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 invariants don't determine.

See pricing

50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

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

More on this topic

The chapter behind this: Classification of second-order PDE — free to read

From Partial Differential Equations › Classification and canonical forms

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