 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-b/5/a/ii/solution

On the real domain

y > 0

M = y (x^{2} + ln y)

and

N = x

have

M_{y} = x^{2} + ln y + 1

and

N_{x} = 1

μ = 1/ y

, which is nonzero on this domain. Multiplication gives

(x^{2} + ln y) d x + \frac{x}{y} d y = d (\frac{x ^{3}}{3} + x ln y) .

\frac{x ^{3}}{3} + x ln y = C, y > 0.

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