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Part BCSIR NET December 2025match-the-characteristic-region-t-less-than-x-or-t-greater-than-x-to-the-right-piece-of-data

Match the characteristic region t less than x or t greater than x to the right piece of data

Consider the initial-boundary value problem (IBVP)ut+ux=0,0<x<∞,t>0,u(x,0)=e) u_{t} + u_{x} = 0, 0 < x < \infty, t > 0, u(x,0) = eˣ, 0<x<∞,u(0,t)=1−sin⁡t,t>00 < x < \infty, u(0,t) = 1 - \sin t, t > 0. Then which of the following statements is true?

  1. A.There exists a unique solution u(x,t) of IBVP such that u(1,t) = e − sin t, for all t < 1.
  2. B.There does NOT exist a solution u(x,t) of IBVP such that u(1,t) = e − sin t, for all t > 1.✓
  3. C.There exists a solution u(x,t) of IBVP such that u(x,t) = e^(x−t), for all t > x.
  4. D.There exists a solution u(x,t) of IBVP such that u(x,t) = e − sin(t−x), for all t < x.

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: A first-order quasilinear Cauchy problem with smooth data has a global smooth solution

More on this topic

The chapter behind this: First-order PDE: characteristics, Lagrange and Charpit — free to read

From Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics

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