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Part CCSIR NET December 2024a-widening-window-adds-logs-a-sliding-one-cancels-them

A widening window adds logs a sliding one cancels them

Let u = u(x, t) be the solution to the initial value problem ∂2u/∂t2−∂2u/∂x2=0,x∈R,t>0,u(x,0)=f(x),x∈R,(∂u/∂t)(x,0)=1/1+x2,x∈R\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = f(x), x \in \mathbb{R}, (\partial{}u/\partial{}t)(x, 0) = 1/\sqrt{1 + x^{2}}, x \in \mathbb{R}, where f is a twice continuously differentiable function defined on R\mathbb{R} satisfying f(x) → 0 as |x| →∞\to \infty. Then which of the following statements are true?

  1. A.For every x∈R,lim⁡(t→∞)u(x,t)=0x \in \mathbb{R}, \lim(t\to\infty) u(x, t) = 0
  2. B.For every x∈R,lim⁡(t→∞)u(x,t)=∞x \in \mathbb{R}, \lim(t\to\infty) u(x, t) = \infty✓ Correct
  3. C.For every t∈(0,∞),lim⁡(x→∞)u(x,t)=0t \in (0, \infty), \lim(x\to\infty) u(x, t) = 0✓ Correct
  4. D.For every t∈(0,∞),lim⁡(x→−∞)u(x,t)=∞t \in (0, \infty), \lim(x\to-\infty) u(x, t) = \infty

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 limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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