The bookUnit 3 · Partial Differential Equations58 / 83

First-order PDE: Lagrange, Charpit, characteristics

Why this is asked: Write the characteristic ODEs, carry the initial data along them, and check whether characteristics cross — that crossing is what makes the solution fail to exist globally.

First-order PDE: characteristics, Lagrange and Charpit

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Lagrange's method on a first-order PDEinteractive

The auxiliary system, two first integrals, and the general solution — worked end to end.

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The trap here

“A first-order quasilinear Cauchy problem with smooth data has a global smooth solution” — false

$u u_{x} + u_y = 0$ with u(x, 0) = −x

Characteristics $x = x_{0}(1 - y)$ all meet at y = 1: the solution blows up in gradient (a shock).