 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/14/a/solution

I (x) = exp \int \frac{x}{a ^{2} + x ^{2}} d x = a^{2} + x^{2} .

\frac{d}{d x} (y a^{2} + x^{2}) = x a^{2} + x^{2} .

The right-hand side integrates to

(a^{2} + x^{2})^{3/2} /3

, so

y = \frac{a ^{2} + x ^{2}}{3} + \frac{C}{a ^{2} + x ^{2}} .

This holds on every interval where the coefficient is defined. If

a = 0

, intervals cannot cross

x = 0

; equivalently the family is

y = x^{2} /3 + C / x

on either half-line, with an independently chosen constant.

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