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ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2015/5/a/i/solution

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Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 5 / a / i / Solution by

Codex  0 2026-10-31

For a second-order linear partial differential equation written

A u_{xx} + 2 B u_{x y} + C u_{yy} + \dots = 0

, the discriminant is

B^{2} - A C

. Here

A = - x^{2}, B = 0, C = 1, B^{2} - A C = x^{2} .

It is positive exactly when

x \neq = 0

, proving that the equation is a hyperbolic partial differential equation on each half-plane. At

x = 0

its principal part degenerates and it is not hyperbolic.

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