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Suppose y = y(x) satisfies the differential equation 8*sqrt(x)*sqrt(9 + sqrt(x)) dy = dx / sqrt(4 + sqrt(9 + sqrt(x))), for x > 0, with y(0) = sqrt(7). Find y(256).
- 80
- 3
- 16
- 9
Correct answer: 3
Solution
Successive substitutions reduce the equation to dy = dw, giving y = sqrt(4 + sqrt(9 + sqrt(x))) + C; applying y(0) = sqrt(7) fixes C = 0, and at x = 256 the value is 3.
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