If y = y (x) satisfies the differential equation 8 x ( 9 + x )dy =...

$y = \dfrac{1}{8}∫ \dfrac{dx}{\sqrt{4 + \sqrt{9 + \sqrt{x}} . \sqrt{x} . \sqrt{9 + \sqrt{x}}}}$

Put $\sqrt{9 + \sqrt{x}} = t ⇒ \dfrac{dx}{\sqrt{x} . \sqrt{9 + \sqrt{x}}} = 4dt$

$∴ y = \dfrac{4}{8} ∫ \dfrac{dt}{\sqrt{4 + t}}$

$⇒ y = \sqrt{4 + t} + C$

$⇒ y\left(x\right) = \sqrt{4 + \sqrt{9 + \sqrt{x}}} + C$

At $x = 0: y\left(0\right) = \sqrt{7} ⇒ C = 0$

$∴ y\left(x\right) = \sqrt{4 + \sqrt{9 + \sqrt{x}}}$

$⇒ y\left(256\right) = 3$