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Let y = y(x) be the solution of the differential equation dy/dx = 2x(x + y)^3 − x(x + y) − 1, y(0) = 1. Then (1/√2 + y(1/√2))^2 equals :

1. 2/(1+√e)
2. 3/(3−√e)
3. 1/(2−√e)
4. 4/(4+√e)

Correct answer: 1/(2−√e)

Solution

The correct option is derived from solving the differential equation and applying the initial condition, which leads to the specific value of y at x = 1/√2. The calculations confirm that (1/√2 + y(1/√2))^2 simplifies to 1/(2−√e), making it the valid choice.

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