 Solution

ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2017/2/b/iii/solution

Two solutions for

x \geq 2

are

y_{1} (x) = - 1, y_{2} (x) = - 1 - \frac{( x - 2 ) ^{2}}{4} .

For the second,

y_{2}^{'} = - (x - 2) /2 = - ∣1 + y_{2} ∣

because

x \geq 2

; both have value

- 1

and derivative zero at two. They are distinct for every

x > 2

. Local Lipschitz continuity fails at the equilibrium, so the existence of these two continuations is compatible with the Picard-Lindelöf theorem.

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