 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-2/5c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

s \mapsto (x (s), y (s))

, the chain rule gives

\frac{d}{d s} u_{x} = u_{xx} \overset{x}{˙} + u_{x y} \overset{y}{˙}, \frac{d}{d s} u_{y} = u_{x y} \overset{x}{˙} + u_{yy} \overset{y}{˙} .

(u_{xx}, u_{x y}, u_{yy})

\overset{x}{˙} 0 a \overset{y}{˙} \overset{x}{˙} 2 b 0 \overset{y}{˙} c .

A characteristic curve is precisely one along which this system is singular, equivalently where the principal symbol vanishes on a normal covector. Taking the determinant yields

a \overset{y}{˙}^{2} - 2 b \overset{x}{˙} \overset{y}{˙} + c \overset{x}{˙}^{2} = 0.

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