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Let y = y(x) satisfy (x² + 1)² dy/dx + 2x(x² + 1) y = 1 with y(0) = 0. If sqrt(a*y(1)) = pi/32, find the value of 32a.

1. pi/4
2. pi/2
3. pi
4. pi/8

Correct answer: pi/4

Solution

The equation becomes d/dx[(x²+1)y] = 1/(x²+1), so (x²+1)y = arctan(x); then y(1)=pi/8 and the condition gives a=pi/128.

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