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Part CCSIR NET December 2023characteristics-crossing

Characteristics crossing

Consider the Cauchy problem u⋅ux+uy=1u\cdot{}u_{x} + u_y = 1 for (x,y)∈R×(0,∞)(x, y) \in \mathbb{R} \times (0, \infty), with u(x, 0) = kx, x∈Rx \in \mathbb{R}, for a real parameter k. For which of the following values of k does the problem have a solution defined on R×(0,∞)\mathbb{R} \times (0, \infty)?

  1. A.k = 0✓ Correct
  2. B.k = −2
  3. C.k = 4✓ Correct
  4. D.k = 1✓ 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 existence vs uniqueness.

See pricing

50 are analysed free — try those first.

The trap it tests

Existence vs uniqueness

A theorem giving one was read as giving both.

Drill statements like this

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

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

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