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