 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2012/9/b/i/solution

x_{s} = 1, y_{s} = z, z_{s} = 2.

Successive integration gives the general characteristic

x = s + a, z = 2 s + c, y = s^{2} + cs + b,

where

a, b, c

are integration constants. Equivalently

z - 2 x

and

y - z^{2} /4

are constant along each curve: differentiating them with respect to

s

gives zero. Three constants occur because shifting the origin of the parameter

s

describes the same geometric curve again; there are two independent curve invariants.

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