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Part BCSIR NET December 2024a-zero-boundary-value-reflects-oddly

A zero boundary value reflects oddly

Let u = u(x, t) be a solution of the wave equation ∂2u/∂t2−∂2u/∂x2=0,x∈R,t>0\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0 satisfying the condition u(0, t) = 0, ∀t ≥ 0. Then which of the following statements is true?

  1. A.u(x, t) = 0, whenever x = t
  2. B.u(x, t) = 0, whenever x = −t
  3. C.u(−x, t) = u(x, t), whenever x > 0, t > 0
  4. D.u(−x, t) = −u(x, t), whenever 0 < x ≤ t✓ 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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: All the classical equations satisfy a maximum principle

More on this topic

The chapter behind this: The three classical equations — free to read

From Partial Differential Equations › Laplace, heat and wave equations: separation of variables

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