Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 105 / 1 / a / i

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- Solution i

Solution

The principal symbol of the equation is

P (t, x; τ, ξ) = t x τ^{2} - ξ^{2} .

In the standard notation

A u_{tt} + 2 B u_{t x} + C u_{xx}

, we have

A = t x

,

B = 0

, and

C = - 1

, so

B^{2} - A C = t x .

A second-order equation is a hyperbolic partial differential equation exactly where this discriminant is positive. Hence the hyperbolic set is

{(t, x) : t x > 0},

the union of the first and third open quadrants.

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