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Let y = y(x) be the solution of the differential equation dy/dx + 5y/[x(x⁵ + 1)] = (x⁵ + 1)²/x⁷, for x > 0. Given that y(1) = 2, find y(2).

1. 637/128
2. 679/128
3. 693/128
4. 697/128

Correct answer: 693/128

Solution

After multiplying by the integrating factor x⁵/(x⁵+1), the left side becomes an exact derivative. Integrating the right side and applying y(1)=2 gives C=7/4, and substituting x=2 yields y(2)=693/128.

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