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ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2021/1/c/i/solution

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Past exam of the differential equations 1 course of the University of Oxford / 2021 / 1 / c / i / Solution by

Codex  0 2026-10-25

In the convention

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

, the coefficients of the principal symbol are

A = x, B = y + \frac{β x}{2}, C = 0, B^{2} - A C = \frac{( 2 y + β x ) ^{2}}{4} .

(1)

Thus the second-order partial differential equation is

hyperbolic, parabolic, 2 y + β x \neq = 0, 2 y + β x = 0, x \neq = 0.

(2)

There is no region of elliptic partial differential equation type. The terms hyperbolic, parabolic and elliptic refer to the pointwise classification of a second-order partial differential equation. At

(0, 0)

every second-order coefficient vanishes, so the principal symbol has rank zero and the equation is degenerate rather than genuinely parabolic there. Moreover the right side is

2

, so no twice continuously differentiable classical solution can satisfy the equation at the origin.

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