2A (Differential Equations)

Since

1 + cos 2 y = 2 cos^{2} y

, separation gives

sec^{2} y d y = \frac{2}{x ^{2}} d x, tan y = C - \frac{2}{x} .

The limiting condition gives

C = 1

. Taking

tan^{- 1}

with range

(- π /2, π /2)

and selecting the branch approaching

9 π /4

yields

y (x) = 2 π + tan^{- 1} (1 - \frac{2}{x})

for

x > 0

b

Write

u = xv

on an interval not containing zero. The equation becomes

x^{2} v^{'} = x

, so

v^{'} = 1/ x

. Hence

u (x) = x {lo g ∣ x ∣ + C} .

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