 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2011/differential-equations/7/ii/solution

No rectangle of positive width around

(0, 0)

y

. For any nonzero

x

in such a rectangle,

\frac{∣ f ( x , y ) - f ( x , 0 ) ∣}{∣ y ∣} = ∣ x ∣ ∣ y ∣^{- 1/3} ⟶ \infty (y \to 0) .

The vanishing factor

x

at the single initial point does not control neighbouring

x

-values. The failure of the hypothesis is accompanied by actual nonuniqueness: both

y_{1} (x) = 0, y_{2} (x) = \frac{x ^{6}}{216}

satisfy the equation and have value zero at

x = 0

. Direct differentiation gives

y_{2}^{'} = x^{5} /36 = x (y_{2})^{2/3}

, including the zero point by continuity.

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