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 $\partial^{2} u / \partial t^{2} - \partial^{2} u / \partial x^{2} = 0, x \in R, t > 0, u (x, 0) = f (x), x \in R, (\partial u / \partial t) (x, 0) = 1/ 1 + x^{2}, x \in R$ , where f is a twice continuously differentiable function defined on $R$ satisfying f(x) → 0 as |x| $\to \infty$ . Then which of the following statements are true?

A.For every $x \in R, lim (t \to \infty) u (x, t) = 0$
B.For every $x \in R, lim (t \to \infty) u (x, t) = \infty$ ✓
C.For every $t \in (0, \infty), lim (x \to \infty) u (x, t) = 0$ ✓
D.For every $t \in (0, \infty), lim (x \to - \infty) u (x, t) = \infty$

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.

See pricing

50 are analysed free — try those first.

Related counterexample: All the classical equations satisfy a maximum principle

A widening window adds logs a sliding one cancels them

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