Question

If 𝑦 = 𝑦(𝑥) satisfies the differential equation
$8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy = \left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1}\,\,dx, \,\,\,\,x > 0$
and $𝑦 (0) = \sqrt{7},$ then $𝑦(256)$ =

• A. 3
• B. 9
• C. 16
• D. 80

Answer 1

Answer: (A)

$dy =\left(\frac{1}{8 \sqrt{ x }(\sqrt{9+\sqrt{ x }})(\sqrt{4+\sqrt{9+\sqrt{ x }}})}\right) dx$
Let $4+\sqrt{9+\sqrt{x}}=t$
$\Rightarrow \frac{1}{2 \sqrt{9+\sqrt{x}}} \cdot \frac{1}{2 \sqrt{x}} d x=d t $
$\Rightarrow d y=\frac{d t}{2 \sqrt{t}}$
$\Rightarrow 2 d y=\frac{1}{\sqrt{t}} d t$
$2 y=2 \sqrt{t}+c$
$\Rightarrow 2 y=2 \sqrt{4+\sqrt{9+\sqrt{x}}}+c$
$y(0)=\sqrt{7} $
$\Rightarrow c=0 $
$y=\sqrt{4+\sqrt{9+\sqrt{x}}}$
$y(256)=3$