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Part BCSIR NET June 2025dalembert-averages-the-displacement-and-integrates-the-velocity-over-the-cone

Dalembert averages the displacement and integrates the velocity over the cone

Let u = u(x, t) be the solution of ∂2u/∂t2−∂2u/∂x2=0,x∈R,t>0,u(x,0)=1+x2,x∈R,∂u/∂t(x,0)=x+1,x∈R\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = 1 + x^{2}, x \in \mathbb{R}, \partial{}u/\partial{}t(x, 0) = x + 1, x \in \mathbb{R}. Then the value of u(1, 1) is

  1. A.2
  2. B.3
  3. C.4
  4. D.5✓ 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 execution slip.

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The trap it tests

Execution slip

The idea was right. The computation was not.

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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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