 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/13/a/solution

I (x) = exp \int \frac{4 x}{1 + x ^{2}} d x = (1 + x^{2})^{2} .

Multiplication converts the left side to a derivative:

\frac{d}{d x} [(1 + x^{2})^{2} y] = \frac{1}{1 + x ^{2}} .

Integrating gives

(1 + x^{2})^{2} y = arctan x + C

. The initial value fixes

C = 1

, so

y (x) = \frac{1 + arctan x}{( 1 + x ^{2} ) ^{2}}, x \in R .

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