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Part BCSIR NET December 2025the-long-time-limit-solves-the-steady-state-ode-not-zero

The long time limit solves the steady state ode not zero

Let u(x,t) be the solution of the partial differential equation ut=16uxx+2,0<x<7,t>0u_{t} = 16u_{xx} + 2, 0 < x < 7, t > 0, satisfying the conditions ux(0,t)=u(7,t)=0u_{x}(0,t) = u(7,t) = 0 for t > 0, and u(x,0) = 0 for 0 < x < 7. Then which of the following statements is true?

  1. A.For each x ∈ (0,7), u(x,t) → 0 as t→∞t \to \infty.
  2. B.For each x∈(0,7),u(x,t)→x2(7−x)2/16x \in (0,7), u(x,t) \to x^{2}(7-x)^{2}/16 as t→∞t \to \infty.
  3. C.For each x∈(0,7),u(x,t)→(x2/16)(7−x)x \in (0,7), u(x,t) \to (x^{2}/16)(7-x) as t→∞t \to \infty.
  4. D.For each x∈(0,7),u(x,t)→(1/16)(49−x2)x \in (0,7), u(x,t) \to (1/16)(49-x^{2}) as t→∞t \to \infty.✓ 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.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

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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