 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/17/b/solution

Away from

x = - 1

\frac{d y}{y} = \frac{x ^{2}}{1 + x ^{3}} d x, ln ∣ y ∣ = \frac{1}{3} ln ∣1 + x^{3} ∣ + C .

On the interval containing

x = 1

, one has

x > - 1

, so the positive initial value selects

y = K (1 + x^{3})^{1/3}

. Substitution of

y (1) = 2

gives

K = 2^{2/3}

. Thus

y (x) = 2^{2/3} (1 + x^{3})^{1/3}, x > - 1.

The zero solution omitted by division by

y

cannot satisfy the initial value. Although the selected solution tends to zero as

x ↓ - 1

, its derivative diverges there; it does not extend through that point as a classical differentiable solution.

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