Skip to content
Part BCSIR NET December 2024the-safe-line-is-where-the-data-vanishes

The safe line is where the data vanishes

The problem ∂u/∂x+∂u/∂y=u2,u(x,0)=x2,∀x∈R\partial{}u/\partial{}x + \partial{}u/\partial{}y = u^{2}, u(x, 0) = x^{2}, \forall{}x \in \mathbb{R} has a solution on an open set containing the line {(x,y)∈R2(x, y) \in \mathbb{R}^{2} : ax + by = 0} if

  1. A.a = 1 and b = 0
  2. B.a = 1 and b = −1✓ Correct
  3. C.a = 2 and b = 1
  4. D.a = 1 and b = 2

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.

ShareWhatsAppTelegram