The problem has a solution on an open set containing the line { : ax + by = 0} if
Part BCSIR NET December 2024the-safe-line-is-where-the-data-vanishes
The safe line is where the data vanishes
Related counterexample: A first-order quasilinear Cauchy problem with smooth data has a global smooth solution
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The chapter behind this: First-order PDE: characteristics, Lagrange and Charpit — free to read
From Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics